The Solution of Linear Simultaneous Equations

Size: px
Start display at page:

Download "The Solution of Linear Simultaneous Equations"

Transcription

1 Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve such a set of equations. The theory of determinants (with applications) can be found in most intermediate-level algebra texts. (A particularly good reference for engineering students is Chapter of E.A. Guillemin s The Mathematics of Circuit Analysis [New York: Wiley, 949]. In our review here, we will limit our discussion to the mechanics of solving simultaneous equations with determinants. A. Preliminary Steps The first step in solving a set of simultaneous equations by determinants is to write the equations in a rectangular (square) format. In other words, we arrange the equations in a vertical stack such that each variable occupies the same horizontal position in every equation. For example, in Eqs. A., the variables i, i 2, i 3 occupy the first, second, third position, respectively, on the left-h side of each equation: 2i - 9i 2-2i 3 = -33, -3i + 6i 2-2i 3 = 3, (A.) -8i - 4i i 3 = 5. Alternatively, one can describe this set of equations by saying that i occupies the first column in the array, i 2 the second column, i 3 the third column. If one or more variables are missing from a given equation, they can be inserted by simply making their coefficient zero. Thus Eqs. A.2 can be squared up as shown by Eqs. A.3: 2v - v 2 = 4, 4v 2 + 3v 3 = 6, (A.2) 7v + 2v 3 = 5; 2v - v 2 + v 3 = 4, v + 4v 2 + 3v 3 = 6, (A.3) 7v + v 2 + 2v 3 = 5. Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. 759

2 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. 76 The Solution of Linear Simultaneous Equations A.2 Cramer s Method The value of each unknown variable in the set of equations is expressed as the ratio of two determinants. If we let N, with an appropriate subscript, represent the numerator determinant represent the denominator determinant, then the kth unknown x k is x k = N k. (A.4) The denominator determinant is the same for every unknown variable is called the characteristic determinant of the set of equations. The numerator determinant N k varies with each unknown. Equation A.4 is referred to as Cramer s method for solving simultaneous equations. A.3 The Characteristic Determinant Once we have organized the set of simultaneous equations into an ordered array, as illustrated by Eqs. A. A.3, it is a simple matter to form the characteristic determinant. This determinant is the square array made up from the coefficients of the unknown variables. For example, the characteristic determinants of Eqs. A. A.3 are = (A.5) 2 - = , 7 2 (A.6) respectively. A.4 The Numerator Determinant The numerator determinant N k is formed from the characteristic determinant by replacing the kth column in the characteristic determinant with the column of values appearing on the right-h side of the equations.

3 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. A.5 The Evaluation of a Determinant 76 For example, the numerator determinants for evaluating,, in Eqs. A. are i i 2 i N = , (A.7) N 2 = , (A.8) N 3 = (A.9) The numerator determinants for the evaluation of,, in Eqs. A.3 are v v 2 v N = , 5 2 (A.) 2 4 N 2 = , (A.) 2-4 N 3 = (A.2) A.5 The Evaluation of a Determinant The value of a determinant is found by exping it in terms of its minors. The minor of any element in a determinant is the determinant that remains after the row column occupied by the element have been deleted. For example, the minor of the element 6 in Eq. A.7 is ,

4 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. 762 The Solution of Linear Simultaneous Equations while the minor of the element 22 in Eq. A.7 is The cofactor of an element is its minor multiplied by the signcontrolling factor where i j denote the row column, respectively, occupied by the element. Thus the cofactor of the element 6 in Eq. A.7 is - (2 + 2) 2 the cofactor of the element 22 is - (3 + 3) 2 The cofactor of an element is also referred to as its signed minor. (i + j) The sign-controlling factor - will equal + or - depending on whether i + j is an even or odd integer. Thus the algebraic sign of a cofactor alternates between + - as we move along a row or column. For a 3 * 3 determinant, the plus minus signs form the checkerboard pattern illustrated here: (i + j), , A determinant can be exped along any row or column. Thus the first step in making an expansion is to select a row i or a column j. Once a row or column has been selected, each element in that row or column is multiplied by its signed minor, or cofactor. The value of the determinant is the sum of these products. As an example, let us evaluate the determinant in Eq. A.5 by exping it along its first column. Following the rules just explained, we write the expansion as =2() (-) () (A.3) The 2 * 2 determinants in Eq. A.3 can also be exped by minors. The minor of an element in a 2 * 2 determinant is a single element. It follows that the expansion reduces to multiplying the upper-left element by the lower-right element then subtracting from this product the product

5 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. A.5 The Evaluation of a Determinant 763 of the lower-left element times the upper-right element. Using this observation, we evaluate Eq. A.3 to =2(32-8) + 3(-98-48) - 8(8 + 72) = = 46. (A.4) Had we elected to exp the determinant along the second row of elements, we would have written =-3(-) (+) (-) = 3(-98-48) + 6(462-96) + 2(-84-72) = = 46. (A.5) The numerical values of the determinants,, given by Eqs. A.7, A.8, A.9 are N N 2 N 3 N = 46, (A.6) N 2 = 2292, (A.7) N 3 = (A.8) i i 2 It follows from Eqs. A.5 through A.8 that the solutions for,, in Eq. A. are i 3 i = N = A, i 2 = N 2 = 2 A, (A.9) i 3 = N 3 = 3 A.

6 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. 764 The Solution of Linear Simultaneous Equations We leave you to verify that the solutions for,, in Eqs. A.3 are v = 49-5 v = -9.8 V, v 2 v 3 v 2 = 8-5 = V, (A.2) v 3 = = 36.8 V. A.6 Matrices A system of simultaneous linear equations can also be solved using matrices. In what follows, we briefly review matrix notation, algebra, terminology. A matrix is by definition a rectangular array of elements; thus a a 2 a 3 Á an a A = D 2 a 22 a 23 Á a2n Á Á Á Á Á T (A.2) a m a m2 a m3 Á amn is a matrix with m rows n columns.we describe A as being a matrix of order m by n, or m * n, where m equals the number of rows n the number of columns.we always specify the rows first the columns second. The elements of the matrix a, a 2, a 3,... can be real numbers, complex numbers, or functions. We denote a matrix with a boldface capital letter. The array in Eq. A.2 is frequently abbreviated by writing A = [a ij ] mn, (A.22) a ij where is the element in the ith row the jth column. If m =, A is called a row matrix, that is, A = [a a 2 a 3 Á an ]. (A.23) An excellent introductory-level text in matrix applications to circuit analysis is Lawrence P. Huelsman, Circuits, Matrices, Linear Vector Spaces (New York: McGraw-Hill, 963).

7 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. A.7 Matrix Algebra 765 If n =, A is called a column matrix, that is, a a 2 A = E a 3 U. o (A.24) If m = n, A is called a square matrix. For example, if m = n = 3, the square 3 by 3 matrix is a m a a 2 a 3 A = C a 2 a 22 a 23 S. a 3 a 32 a 33 (A.25) Also note that we use brackets [] to denote a matrix, whereas we use vertical lines ƒƒ to denote a determinant. It is important to know the difference. A matrix is a rectangular array of elements. A determinant is a function of a square array of elements.thus if a matrix A is square, we can define the determinant of A. For example, if then A = c d, det A = = 3-6 = 24. A.7 Matrix Algebra The equality, addition, subtraction of matrices apply only to matrices of the same order. Two matrices are equal if, only if, their corresponding elements are equal. In other words, A = B if, only if, a ij = b ij for all i j. For example, the two matrices in Eqs. A.26 A.27 are equal because a = b, a 2 = b 2, a 2 = b 2, a 22 = b 22 : 36-2 A = c 4 6 d, (A.26) 36-2 B = c 4 6 d. (A.27)

8 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. 766 The Solution of Linear Simultaneous Equations If A B are of the same order, then C = A + B (A.28) implies c ij = a ij + b ij. (A.29) For example, if 4-6 A = c d, (A.3) 6-3 B = c d, (A.3) then C = c -2 2 d. (A.32) The equation D = A - B (A.33) implies d ij = a ij - b ij. (A.34) For the matrices in Eqs. A.3 A.3, we would have D = c d. (A.35) Matrices of the same order are said to be conformable for addition subtraction. Multiplying a matrix by a scalar k is equivalent to multiplying each element by the scalar. Thus A = kb if, only if, a ij = kb ij. It should be noted that k may be real or complex. As an example, we will multiply the matrix D in Eq. A.35 by 5. The result is D = c d. (A.36)

9 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. A.7 Matrix Algebra 767 Matrix multiplication can be performed only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In other words, the product AB requires the number of columns in A to equal the number of rows in B. The order of the resulting matrix will be the number of rows in A by the number of columns in B. Thus if C = AB, where A is of order m * p B is of order p * n, then C will be a matrix of order m * n. When the number of columns in A equals the number of rows in B, we say A is conformable to B for multiplication. An element in C is given by the formula c ij = a p k = a ik b kj. (A.37) The formula given by Eq. A.37 is easy to use if one remembers that matrix multiplication is a row-by-column operation. Hence to get the ith, jth term in C, each element in the ith row of A is multiplied by the corresponding element in the jth column of B, the resulting products are summed. The following example illustrates the procedure. We are asked to find the matrix C when A = c d (A.38) 4 2 B = C 3S. -2 (A.39) First we note that C will be a 2 * 2 matrix that each element in C will require summing three products. To find C we multiply the corresponding elements in row of matrix A with the elements in column of matrix B then sum the products. We can visualize this multiplication summing process by extracting the corresponding row column from each matrix then lining them up element by element. So to find we have therefore C Row of A Column of B ; C = 6 * * + 2 * = 26. To find C 2 we visualize Row of A Column 2 of B ;

10 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. 768 The Solution of Linear Simultaneous Equations thus C 2 = 6 * * * (-2) = 7. For C 2 we have Row 2 of A Column of B ; C 2 = * * + 6 * =. Finally, for C 22 we have Row 2 of A Column 2 of B ; from which C 22 = * * * (-2) = 2. It follows that 26 7 C = AB = B 2 R. (A.4) In general, matrix multiplication is not commutative, that is, AB Z BA. As an example, consider the product BA for the matrices in Eqs. A.38 A.39. The matrix generated by this multiplication is of order 3 * 3, each term in the resulting matrix requires adding two products. Therefore if D = BA, we have D = C 3 2 8S (A.4) Obviously, C Z D. We leave you to verify the elements in Eq. A.4. Matrix multiplication is associative distributive. Thus (AB)C = A(BC), (A.42) A(B + C) = AB + AC, (A.43) (A + B)C = AC + BC. (A.44)

11 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. A.7 Matrix Algebra 769 In Eqs. A.42, A.43, A.44, we assume that the matrices are conformable for addition multiplication. We have already noted that matrix multiplication is not commutative. There are two other properties of multiplication in scalar algebra that do not carry over to matrix algebra. First, the matrix product AB = does not imply either A = or B =.(Note: A matrix is equal to zero when all its elements are zero.) For example, if then A = c d B = c d, Hence the product is zero, but neither A nor B is zero. Second, the matrix equation AB = AC does not imply B = C. For example, if then AB = c d =. A = c 4 4 d, B = c3 d, C = c d, AB = AC = c 3 4 d, but B Z C. 6 8 The transpose of a matrix is formed by interchanging the rows columns. For example, if A = C 4 5 6S, then A T = C 2 5 8S The transpose of the sum of two matrices is equal to the sum of the transposes, that is, (A + B) T = A T + B T. (A.45) The transpose of the product of two matrices is equal to the product of the transposes taken in reverse order. In other words, [AB] T = B T A T. (A.46)

12 77 The Solution of Linear Simultaneous Equations Equation A.46 can be extended to a product of any number of matrices. For example, [ABCD] T = D T C T B T A T. (A.47) If A = A T, the matrix is said to be symmetric. Only square matrices can be symmetric. A.8 Identity, Adjoint, Inverse Matrices An identity matrix is a square matrix where a ij = for i Z j, a ij = for i = j. In other words, all the elements in an identity matrix are zero except those along the main diagonal, where they are equal to. Thus c d, C S, D T are all identity matrices. Note that identity matrices are always square. We will use the symbol U for an identity matrix. The adjoint of a matrix A of order n * n is defined as adj A = [ ji ] n * n, (A.48) ij a ij where is the cofactor of. (See Section A.5 for the definition of a cofactor.) It follows from Eq. A.48 that one can think of finding the adjoint of a square matrix as a two-step process. First construct a matrix made up of the cofactors of A, then transpose the matrix of cofactors. As an example we will find the adjoint of the 3 * 3 matrix 2 3 A = C 3 2 S. - 5 The cofactors of the elements in A are = ( - ) = 9, 2 = -(5 + ) = -6, 3 = (3 + 2) = 5, 2 = -( - 3) = -7, 22 = (5 + 3) = 8, 23 = -( + 2) = -3, 3 = (2-6) = -4, 32 = -( - 9) = 8, 33 = (2-6) = -4. Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel.

13 A.8 Identity, Adjoint, Inverse Matrices 77 The matrix of cofactors is It follows that the adjoint of A is B = C S adj A = B T = C S One can check the arithmetic of finding the adjoint of a matrix by using the theorem adj A # A = det A # U. (A.49) Equation A.49 tells us that the adjoint of A times A equals the determinant of A times the identity matrix, or for our example, det A = (9) + 3(-7) - (-4) = -8. If we let C = adj A # A use the technique illustrated in Section A.7, we find the elements of C to be c = = -8, c 2 = =, c 3 = =, c 2 = =, c 22 = = -8, c 23 = =, c 3 = =, c 32 = =, c 33 = = -8. Therefore -8 C = C -8 S = -8C S -8 = det A # U. A square matrix A has an inverse, denoted as A - A = AA - = U. A -, if Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. (A.5)

14 2 You can learn alternative methods for finding the inverse in any introductory text on matrix theory. See, for example, Franz E. Hohn, Elementary Matrix Algebra (New York: Macmillan, 973). Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. 772 The Solution of Linear Simultaneous Equations Equation A.5 tells us that a matrix either premultiplied or postmultiplied by its inverse generates the identity matrix U. For the inverse matrix to exist, it is necessary that the determinant of A not equal zero. Only square matrices have inverses, the inverse is also square. A formula for finding the inverse of a matrix is A - = adj A det A. (A.5) The formula in Eq. A.5 becomes very cumbersome if A is of an order larger than 3 by 3. 2 Today the digital computer eliminates the drudgery of having to find the inverse of a matrix in numerical applications of matrix algebra. It follows from Eq. A.5 that the inverse of the matrix A in the previous example is A - = ->8C S = C S You should verify that A - A = AA - = U. A.9 Partitioned Matrices It is often convenient in matrix manipulations to partition a given matrix into submatrices. The original algebraic operations are then carried out in terms of the submatrices. In partitioning a matrix, the placement of the partitions is completely arbitrary, with the one restriction that a partition must dissect the entire matrix. In selecting the partitions, it is also necessary to make sure the submatrices are conformable to the mathematical operations in which they are involved. For example, consider using submatrices to find the product C = AB, where A = E U - 2-2

15 A.9 Partitioned Matrices B = E - U. 3 Assume that we decide to partition B into two submatrices, B B 2 ; thus Now since B has been partitioned into a two-row column matrix, A must be partitioned into at least a two-column matrix; otherwise the multiplication cannot be performed. The location of the vertical partitions of the A matrix will depend on the definitions of. For example, if A B = c B B 2 d. B 2 B = C S B 2 = c 3 d, - then must contain three columns, must contain two columns. Thus the partitioning shown in Eq. A.52 would be acceptable for executing the product AB: B 2 A C = E U F Á 3 V. (A.52) If, on the other h, we partition the B matrix so that A B = c 2 - d B 2 = C 3 S, then must contain two columns, must contain three columns. In this case the partitioning shown in Eq.A.53 would be acceptable in executing the product C = AB: A Á C = E U F V. (A.53) Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel.

16 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. 774 The Solution of Linear Simultaneous Equations For purposes of discussion, we will focus on the partitioning given in Eq. A.52 leave you to verify that the partitioning in Eq. A.53 leads to the same result. From Eq. A.52 we can write C = [A A 2 ] c B B 2 d = A B + A 2 B 2. (A.54) It follows from Eqs. A.52 A.54 that A B = E - 2U C S = E -4 U, A 2 B 2 = E -3 U c 3 6 d = E -9 U, C = E -3 U. -7 The A matrix could also be partitioned horizontally once the vertical partitioning is made consistent with the multiplication operation. In this simple problem, the horizontal partitions can be made at the discretion of the analyst. Therefore C could also be evaluated using the partitioning shown in Eq. A.55: Á C = F Á Á Á Á Á V F Á 3 V. (A.55)

17 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. A.9 Partitioned Matrices 775 From Eq. A.55 it follows that C = c A A 2 A 2 A 22 d c B B 2 d = c C C 2 d, (A.56) where C = A B + A 2 B 2, C 2 = A 2 B + A 22 B 2. You should verify that C = c d C S + c d c3 d - = c - 7 d + c2 6 d = c 3 d, C 2 = C - S C S + C S c 3 d = C S + C S = C S, C = E -3 U. -7 We note in passing that the partitioning in Eqs. A.52 A.55 is conformable with respect to addition.

18 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. 776 The Solution of Linear Simultaneous Equations A. Applications The following examples demonstrate some applications of matrix algebra in circuit analysis. Example A. Use the matrix method to solve for the node voltages in Eqs Solution The first step is to rewrite Eqs in matrix notation. Collecting the coefficients of v v 2 at the same time shifting the constant terms to the right-h side of the equations gives us (A.57) It follows that in matrix notation, Eq. A.57 becomes or where (A.58) (A.59) To find the elements of the V matrix, we premultiply both sides of Eq. A.59 by the inverse of A; thus or v v 2.7v -.5v 2 =, -.5v +.6v 2 = c d cv d = c v 2 2 d, AV = I, A = c d, V = c v v 2 d, I = c 2 d. A - AV = A - I. Equation A.6 reduces to UV = A - I, V = A - I. (A.6) (A.6) (A.62) It follows from Eq. A.62 that the solutions for v v 2 are obtained by solving for the matrix product A - I. To find the inverse of A, we first find the cofactors of A. Thus The matrix of cofactors is the adjoint of A is The determinant of A is det A = 2 = (-) 2 (.6) =.6, 2 = (-) 3 (-.5) =.5, 2 = (-) 3 (-.5) =.5, 22 = (-) 4 (.7) = B = c.5.7 d, adj A = B T.6.5 = c.5.7 d. (A.63) (A.64) (A.65) = (.7)(.6) - (.25) =.77. (A.66) From Eqs. A.65 A.66, we can write the inverse of the coefficient matrix, that is, Now the product A - I is found: It follows directly that A - =.5 c d. A - I = 77 c d c 2 d = 77 c d = c d. c v 9.9 d = c v 2.9 d, or v = 9.9 V v 2 =.9 V. (A.67) (A.68) (A.69)

19 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. A. Applications 777 Example A.2 Use the matrix method to find the three mesh currents in the circuit in Fig Solution The mesh-current equations that describe the circuit in Fig are given in Eq The constraint equation imposed by the current-controlled voltage source is given in Eq When Eq is substituted into Eq. 4.34, the following set of equations evolves: In matrix notation, Eqs. A.7 reduce to where 25i i - 5i 2-2i 3 = 5, -5i i + i 2-4i 3 =, -5i - 4i 2 + 9i 3 = A = C -5-4 S, i I = C i 2 S, i 3 AI = V, 5 V = C S. (A.7) (A.7) It follows from Eq.A.7 that the solution for I is I = A - V. (A.72) We find the inverse of A by using the relationship A - = adj A det A. (A.73) To find the adjoint of A, we first calculate the cofactors of A. Thus = (-) 2 (9-6) = 74, 2 = (-) 3 (-45-2) = 65, 3 = (-) 4 (2 + 5) = 7, 2 = (-) 3 (-45-8) = 25, 22 = (-) 4 (225 - ) = 25, 23 = (-) 5 (- - 25) = 25, 3 = (-) 4 (2 + 2) = 22, 32 = (-) 5 (- - ) = 2, 33 = (-) 6 (25-25) = 225. The cofactor matrix is from which we can write the adjoint of A: adj A = B T = C S The determinant of A is det A = B = C S, It follows from Eq. A.73 that The solution for I is A - = C S (A.74) (A.75) = 25(9-6) + 5(-45-8) - 5(2 + 2) = 25. (A.76) I = (A.77) 25 C S C S = C 26. S The mesh currents follow directly from Eq.A.77.Thus i i 29.6 C i 2 S = C 26. S i (A.78) or i = 29.6 A, i 2 = 26 A, i 3 = 28 A. Example A.3 illustrates the application of the matrix method when the elements of the matrix are complex numbers.

20 778 The Solution of Linear Simultaneous Equations Example A.2 A.3 Use the matrix method to find the phasor mesh currents in the circuit in Fig Solution Summing the voltages around mesh generates the equation (A.79) Summing the voltages around mesh 2 produces the equation (2 - j6)(i 2 - I ) + ( + j3)i I x =. (A.8) The current controlling the dependent voltage source is (A.8) After substituting Eq. A.8 into Eq. A.8, the equations are put into a matrix format by first collecting, in each equation, the coefficients of ; thus (3 - j4)i - (2 - j6)i 2 = 5l, (27 + j6)i - (26 + j3)i 2 =. Now, using matrix notation, Eq. A.82 is written where I I 2 ( + j2)i + (2 - j6)(i - I 2 ) = 5l. I x = (I - I 2 ). AI = V, 3 - j4 -(2 - j6) A = c 27 + j6 -(26 + j3) d, I = c I d, V = c 5l d. I 2 It follows from Eq. A.83 that I = A - V. (A.82) (A.83) (A.84) The inverse of the coefficient matrix A is found using Eq. A.73. In this case, the cofactors of A are = (-) 2 (-26 - j3) = j3, 2 = (-) 3 (27 + j6) = j6, 2 = (-) 3 (-2 + j6) = 2 - j6, 22 = (-) 4 (3 - j4) = 3 - j4. I I 2 The cofactor matrix B is (-26 - j3) (-27 - j6) B = c (2 - j6) (3 - j4) d. The adjoint of A is adj A = B T (-26 - j3) (2 - j6) = c (-27 - j6) (3 - j4) d. The determinant of A is det A = 2 (3 - j4) -(2 - j6) (27 + j6) -(26 + j3) 2 = 6 - j45. (A.85) (A.86) = -(3 - j4)(26 + j3) + (2 - j6)(27 + j6) The inverse of the coefficient matrix is Equation A.88 can be simplified to A - = (-26 - j3) (2 - j6) c A - (-27 - j6) (3 - j4) d =. (6 - j45) 6 + j Substituting Eq. A.89 into A.84 gives us (-26 - j52) = c (-24 - j58) d. It follows from Eq. A.9 that (-26 - j3) (2 - j6) c (-27 - j6) (3 - j4) d = j j28 c j j7 d. I = (-26 - j52) = 58.4l A, I 2 = (-24 - j58) = 62.77l A. (A.87) (A.88) (A.89) c I d = - j3) (96 - j28) c(-65 I (-6 - j45) (94 - j7) d c5l d (A.9) (A.9) In the first three examples, the matrix elements have been numbers real numbers in Examples A. A.2, complex numbers in Example A.3. It is also possible for the elements to be functions. Example A.4 illustrates the use of matrix algebra in a circuit problem where the elements in the coefficient matrix are functions. Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel.

21 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. A. Applications 779 Example A.4 Use the matrix method to derive expressions for the node voltages in the circuit in Fig. A.. Solution Summing the currents away from nodes 2 generates the following set of equations: Letting G = >R collecting the coefficients of V gives us Writing Eq. A.93 in matrix notation yields where V - V g + V R sc + (V - V 2 )sc =, (A.92) V 2 R + (V 2 - V )sc + (V 2 - V g )sc =. V 2 V -scv + (G + 2sC)V 2 = scv g. A = c G + 2sC -sc -sc G + 2sC d, V = c V V 2 d, I = c GV g scv g d. It follows from Eq. A.94 that (A.93) (A.94) (A.95) As before, we find the inverse of the coefficient matrix by first finding the adjoint of A the determinant of A. The cofactors of A are = (-) 2 [G + 2sC] = G + 2sC, 2 = (-) 3 (-sc) = sc, 2 = (-) 3 (-sc) = sc, 22 = (-) 4 [G + 2sC] = G + 2sC. The cofactor matrix is V 2 (G + 2sC)V - scv 2 = GV g, AV = I, V = A - I. B = c G + 2sC sc sc G + 2sC d, (A.96) therefore the adjoint of the coefficient matrix is adj A = B T = c G + 2sC sc (A.97) sc G + 2sC d. R v g v The determinant of A is sc det A = 2 G + 2sC sc sc G + 2sC 2 = G 2 + 4sCG + 3s 2 C 2. sc Figure A. The circuit for Example A.4. sc The inverse of the coefficient matrix is A - = It follows from Eq. A.95 that c G + 2sC sc sc G + 2sC d (G 2 + 4sCG + 3s 2 C 2 ). c V c G + 2sC sc sc G + 2sC d c GV g d scv g d = V 2 (G 2 + 4sCG + 3s 2 C 2 ) v 2 R (A.98) (A.99) (A.) Carrying out the matrix multiplication called for in Eq. A. gives c V V 2 d = (G 2 + 4sCG + 3s 2 C 2 ) c(g2 + 2sCG + s 2 C 2 )V g (2sCG + 2s 2 C 2 )V g d.. (A.) Now the expressions for V V 2 can be written directly from Eq. A.; thus V = (G2 + 2sCG + s 2 C 2 )V g (G 2 + 4sCG + 3s 2 C 2 ), V 2 = 2(sCG + s 2 C 2 )V g (G 2 + 4sCG + 3s 2 C 2 ). (A.2) (A.3)

22 Electric Circuits, Eighth Edition, by James A. Nilsson Susan A. Riedel. 78 The Solution of Linear Simultaneous Equations In our final example, we illustrate how matrix algebra can be used to analyze the cascade connection of two two-port circuits. Example A.5 Show by means of matrix algebra how the input variables V I can be described as functions of the output variables V 2 I 2 in the cascade connection shown in Fig. 8.. Solution We begin by expressing, in matrix notation, the relationship between the input output variables of each two-port circuit. Thus c V d = c a I c V fl I d = ca fl (A.4) (A.5) Now the cascade connection imposes the constraints V 2 = V a 2 a 2 -a 2 -a 22 fl -a 2 -a 22 I 2 d c V 2 I 2 d, fl d cv 2 d, I 2 = -I. (A.6) These constraint relationships are substituted into Eq. A.4. Thus c V d = c a I (A.7) The relationship between the input variables (, I ) the output variables ( V 2, I 2 ) is obtained by substituting Eq. A.5 into Eq. A.7. The result is c V d = c a I a 2 a 2 = c a a 2 a 2 a 2 a 22 -a 2 -a 22 a 22 d c a fl fl fl -a 2 a 2 d c -I d d c V I d. fl -a d cv 2 d. 22 I 2 (A.8) After multiplying the coefficient matrices, we have V c V d = c (a a fl + a 2 a fl 2 ) -(a a fl 2 + a 2 I (a 2 a fl + a 22 a fl 2 ) -(a 2 a fl 2 + a 22 V a fl 22 ) a fl 22 ) d cv 2 (A.9) I 2 d. Note that Eq. A.9 corresponds to writing Eqs in matrix form.

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

4. MATRICES Matrices

4. MATRICES Matrices 4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

More information

9 Matrices, determinants, inverse matrix, Cramer s Rule

9 Matrices, determinants, inverse matrix, Cramer s Rule AAC - Business Mathematics I Lecture #9, December 15, 2007 Katarína Kálovcová 9 Matrices, determinants, inverse matrix, Cramer s Rule Basic properties of matrices: Example: Addition properties: Associative:

More information

( % . This matrix consists of $ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&

( % . This matrix consists of $ 4 5  5' the coefficients of the variables as they appear in the original system. The augmented 3  2 2 # 2  3 4& Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important

More information

Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you: Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants

More information

= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are

= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In

More information

Mathematics Notes for Class 12 chapter 3. Matrices

Mathematics Notes for Class 12 chapter 3. Matrices 1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form

More information

The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23

The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23 (copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional

More information

(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

More information

Introduction to Matrix Algebra I

Introduction to Matrix Algebra I Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model

More information

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11

More information

Chapter 4: Binary Operations and Relations

Chapter 4: Binary Operations and Relations c Dr Oksana Shatalov, Fall 2014 1 Chapter 4: Binary Operations and Relations 4.1: Binary Operations DEFINITION 1. A binary operation on a nonempty set A is a function from A A to A. Addition, subtraction,

More information

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

More information

Appendix B: Solving Simultaneous Equations

Appendix B: Solving Simultaneous Equations Appendix B: Solving Simultaneous Equations Electric circuit analysis methods help us solve for the unknown voltages and currents in the circuits: Kirchhoff s circuit law solves for unknown branch currents

More information

EC9A0: Pre-sessional Advanced Mathematics Course

EC9A0: Pre-sessional Advanced Mathematics Course University of Warwick, EC9A0: Pre-sessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Pre-sessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond

More information

1 Vector Spaces and Matrix Notation

1 Vector Spaces and Matrix Notation 1 Vector Spaces and Matrix Notation De nition 1 A matrix: is rectangular array of numbers with n rows and m columns. 1 1 1 a11 a Example 1 a. b. c. 1 0 0 a 1 a The rst is square with n = and m = ; the

More information

Diagonal, Symmetric and Triangular Matrices

Diagonal, Symmetric and Triangular Matrices Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

More information

The Characteristic Polynomial

The Characteristic Polynomial Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

More information

MATH36001 Background Material 2015

MATH36001 Background Material 2015 MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be

More information

Matrix Algebra and Applications

Matrix Algebra and Applications Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2 - Matrices and Matrix Algebra Reading 1 Chapters

More information

1 Determinants. Definition 1

1 Determinants. Definition 1 Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described

More information

Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

More information

1 Introduction to Matrices

1 Introduction to Matrices 1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

More information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,

Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n, LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x

More information

Introduction to Matrix Algebra

Introduction to Matrix Algebra Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

More information

Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants

Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Hartmut Führ fuehr@matha.rwth-aachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview

More information

MATHEMATICS FOR ENGINEERS BASIC MATRIX THEORY TUTORIAL 2

MATHEMATICS FOR ENGINEERS BASIC MATRIX THEORY TUTORIAL 2 MATHEMATICS FO ENGINEES BASIC MATIX THEOY TUTOIAL This is the second of two tutorials on matrix theory. On completion you should be able to do the following. Explain the general method for solving simultaneous

More information

Chapter 8. Matrices II: inverses. 8.1 What is an inverse?

Chapter 8. Matrices II: inverses. 8.1 What is an inverse? Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we

More information

Direct Methods for Solving Linear Systems. Linear Systems of Equations

Direct Methods for Solving Linear Systems. Linear Systems of Equations Direct Methods for Solving Linear Systems Linear Systems of Equations Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University

More information

Lecture 2 Matrix Operations

Lecture 2 Matrix Operations Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or

More information

Notes on Determinant

Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

More information

Determinants. Dr. Doreen De Leon Math 152, Fall 2015

Determinants. Dr. Doreen De Leon Math 152, Fall 2015 Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.

More information

A Brief Primer on Matrix Algebra

A Brief Primer on Matrix Algebra A Brief Primer on Matrix Algebra A matrix is a rectangular array of numbers whose individual entries are called elements. Each horizontal array of elements is called a row, while each vertical array is

More information

Chapter 1 - Matrices & Determinants

Chapter 1 - Matrices & Determinants Chapter 1 - Matrices & Determinants Arthur Cayley (August 16, 1821 - January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed

More information

Basics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20

Basics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20 Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical

More information

Solving a System of Equations

Solving a System of Equations 11 Solving a System of Equations 11-1 Introduction The previous chapter has shown how to solve an algebraic equation with one variable. However, sometimes there is more than one unknown that must be determined

More information

Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses

Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,

More information

Matrix Inverse and Determinants

Matrix Inverse and Determinants DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and

More information

DETERMINANTS. b 2. x 2

DETERMINANTS. b 2. x 2 DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

More information

2.5 Elementary Row Operations and the Determinant

2.5 Elementary Row Operations and the Determinant 2.5 Elementary Row Operations and the Determinant Recall: Let A be a 2 2 matrtix : A = a b. The determinant of A, denoted by det(a) c d or A, is the number ad bc. So for example if A = 2 4, det(a) = 2(5)

More information

Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

More information

In this leaflet we explain what is meant by an inverse matrix and how it is calculated.

In this leaflet we explain what is meant by an inverse matrix and how it is calculated. 5.5 Introduction The inverse of a matrix In this leaflet we explain what is meant by an inverse matrix and how it is calculated. 1. The inverse of a matrix The inverse of a square n n matrix A, is another

More information

Matrix Differentiation

Matrix Differentiation 1 Introduction Matrix Differentiation ( and some other stuff ) Randal J. Barnes Department of Civil Engineering, University of Minnesota Minneapolis, Minnesota, USA Throughout this presentation I have

More information

1.5 Elementary Matrices and a Method for Finding the Inverse

1.5 Elementary Matrices and a Method for Finding the Inverse .5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:

More information

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions. 3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

More information

We know a formula for and some properties of the determinant. Now we see how the determinant can be used.

We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we

More information

Inverses and powers: Rules of Matrix Arithmetic

Inverses and powers: Rules of Matrix Arithmetic Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

More information

Introduction to Matrices for Engineers

Introduction to Matrices for Engineers Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 0-8 4 0-1 1 0 11

More information

2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors

2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors 2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the

More information

A matrix over a field F is a rectangular array of elements from F. The symbol

A matrix over a field F is a rectangular array of elements from F. The symbol Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

More information

Mathematics of Cryptography

Mathematics of Cryptography CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

More information

The Inverse of a Matrix

The Inverse of a Matrix The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square

More information

Undergraduate Matrix Theory. Linear Algebra

Undergraduate Matrix Theory. Linear Algebra Undergraduate Matrix Theory and Linear Algebra a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn John S Alin Linfield College Colin L Starr Willamette University December 15, 2015 ii Contents 1 SYSTEMS OF LINEAR

More information

Matrices, transposes, and inverses

Matrices, transposes, and inverses Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrix-vector multiplication: two views st perspective: A x is linear combination of columns of A 2 4

More information

Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6

Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6 Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a

More information

APPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the co-factor matrix [A ij ] of A.

APPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the co-factor matrix [A ij ] of A. APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the co-factor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj

More information

Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices

Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices Solving square systems of linear equations; inverse matrices. Linear algebra is essentially about solving systems of linear equations,

More information

1 Introduction. 2 Matrices: Definition. Matrix Algebra. Hervé Abdi Lynne J. Williams

1 Introduction. 2 Matrices: Definition. Matrix Algebra. Hervé Abdi Lynne J. Williams In Neil Salkind (Ed.), Encyclopedia of Research Design. Thousand Oaks, CA: Sage. 00 Matrix Algebra Hervé Abdi Lynne J. Williams Introduction Sylvester developed the modern concept of matrices in the 9th

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

A general system of m equations in n unknowns. a 11 x 1 + a 12 x a 1n x n = a 21 x 1 + a 22 x a 2n x n =

A general system of m equations in n unknowns. a 11 x 1 + a 12 x a 1n x n = a 21 x 1 + a 22 x a 2n x n = A general system of m equations in n unknowns MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) a 11 x 1 + a 12 x 2 + + a 1n x n a 21 x 1

More information

Linear Algebra: Matrices

Linear Algebra: Matrices B Linear Algebra: Matrices B 1 Appendix B: LINEAR ALGEBRA: MATRICES TABLE OF CONTENTS Page B.1. Matrices B 3 B.1.1. Concept................... B 3 B.1.2. Real and Complex Matrices............ B 3 B.1.3.

More information

Homework: 2.1 (page 56): 7, 9, 13, 15, 17, 25, 27, 35, 37, 41, 46, 49, 67

Homework: 2.1 (page 56): 7, 9, 13, 15, 17, 25, 27, 35, 37, 41, 46, 49, 67 Chapter Matrices Operations with Matrices Homework: (page 56):, 9, 3, 5,, 5,, 35, 3, 4, 46, 49, 6 Main points in this section: We define a few concept regarding matrices This would include addition of

More information

Matrices and Systems of Linear Equations

Matrices and Systems of Linear Equations Chapter Matrices and Systems of Linear Equations In Chapter we discuss how to solve a system of linear equations. If there are not too many equations or unknowns our task is not very difficult; what we

More information

Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.

Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that

More information

MAT188H1S Lec0101 Burbulla

MAT188H1S Lec0101 Burbulla Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u

More information

10. Graph Matrices Incidence Matrix

10. Graph Matrices Incidence Matrix 10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

More information

NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

More information

Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations. Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

More information

L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014

L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 Unfortunately, no one can be told what the Matrix is. You have to see it for yourself. -- Morpheus Primary concepts:

More information

GRA6035 Mathematics. Eivind Eriksen and Trond S. Gustavsen. Department of Economics

GRA6035 Mathematics. Eivind Eriksen and Trond S. Gustavsen. Department of Economics GRA635 Mathematics Eivind Eriksen and Trond S. Gustavsen Department of Economics c Eivind Eriksen, Trond S. Gustavsen. Edition. Edition Students enrolled in the course GRA635 Mathematics for the academic

More information

SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison

SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections

More information

Matrix Solution of Equations

Matrix Solution of Equations Contents 8 Matrix Solution of Equations 8.1 Solution by Cramer s Rule 2 8.2 Solution by Inverse Matrix Method 13 8.3 Solution by Gauss Elimination 22 Learning outcomes In this Workbook you will learn to

More information

Matrices Worksheet. Adding the results together, using the matrices, gives

Matrices Worksheet. Adding the results together, using the matrices, gives Matrices Worksheet This worksheet is designed to help you increase your confidence in handling MATRICES. This worksheet contains both theory and exercises which cover. Introduction. Order, Addition and

More information

Math 315: Linear Algebra Solutions to Midterm Exam I

Math 315: Linear Algebra Solutions to Midterm Exam I Math 35: Linear Algebra s to Midterm Exam I # Consider the following two systems of linear equations (I) ax + by = k cx + dy = l (II) ax + by = 0 cx + dy = 0 (a) Prove: If x = x, y = y and x = x 2, y =

More information

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

More information

Sergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014

Sergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014 Sergei Silvestrov,, Karl Lundengård, Johan Richter, Jonas Österberg November 13, 2014 Analysis Todays lecture: Course overview. Repetition of matrices elementary operations. Repetition of solvability of

More information

row row row 4

row row row 4 13 Matrices The following notes came from Foundation mathematics (MATH 123) Although matrices are not part of what would normally be considered foundation mathematics, they are one of the first topics

More information

Solving Systems of Linear Equations. Substitution

Solving Systems of Linear Equations. Substitution Solving Systems of Linear Equations There are two basic methods we will use to solve systems of linear equations: Substitution Elimination We will describe each for a system of two equations in two unknowns,

More information

Chapter 1 Linear Equations and Matrices

Chapter 1 Linear Equations and Matrices Chapter : Linear Equations and Matrices Chapter Linear Equations and Matrices SECION A Systems of Linear Equations By the end of this section you will be able to understand what is meant by a linear system

More information

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics

More information

Economics 102C: Advanced Topics in Econometrics 2 - Tooling Up: The Basics

Economics 102C: Advanced Topics in Econometrics 2 - Tooling Up: The Basics Economics 102C: Advanced Topics in Econometrics 2 - Tooling Up: The Basics Michael Best Spring 2015 Outline The Evaluation Problem Matrix Algebra Matrix Calculus OLS With Matrices The Evaluation Problem:

More information

B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix

B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.

More information

Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that

Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that 0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c

More information

5.3 Determinants and Cramer s Rule

5.3 Determinants and Cramer s Rule 290 5.3 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given

More information

Facts About Eigenvalues

Facts About Eigenvalues Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v

More information

Name: Section Registered In:

Name: Section Registered In: Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

More information

MATH 240 Fall, Chapter 1: Linear Equations and Matrices

MATH 240 Fall, Chapter 1: Linear Equations and Matrices MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS

More information

Further Maths Matrix Summary

Further Maths Matrix Summary Further Maths Matrix Summary A matrix is a rectangular array of numbers arranged in rows and columns. The numbers in a matrix are called the elements of the matrix. The order of a matrix is the number

More information

The Cayley Hamilton Theorem

The Cayley Hamilton Theorem The Cayley Hamilton Theorem Attila Máté Brooklyn College of the City University of New York March 23, 2016 Contents 1 Introduction 1 1.1 A multivariate polynomial zero on all integers is identically zero............

More information

Typical Linear Equation Set and Corresponding Matrices

Typical Linear Equation Set and Corresponding Matrices EWE: Engineering With Excel Larsen Page 1 4. Matrix Operations in Excel. Matrix Manipulations: Vectors, Matrices, and Arrays. How Excel Handles Matrix Math. Basic Matrix Operations. Solving Systems of

More information

NOTES on LINEAR ALGEBRA 1

NOTES on LINEAR ALGEBRA 1 School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura

More information

The Inverse of a Square Matrix

The Inverse of a Square Matrix These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

More information

MATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.

MATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada. MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division.

More information

Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013

Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013 Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous

More information

A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form

A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...

More information

Determinants. Chapter Properties of the Determinant

Determinants. Chapter Properties of the Determinant Chapter 4 Determinants Chapter 3 entailed a discussion of linear transformations and how to identify them with matrices. When we study a particular linear transformation we would like its matrix representation

More information