# 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

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1 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 a 1n x n b 1 a 21 x 1 + a 22 x a 2n x n b 2 a 31 x 1 + a 32 x a 3n x n b (1) a m1 x 1 + a m2 x a mn x n b m In this system, the a ij s and b i s are given real numbers; a ij is the coefficient for the unknown x j in the ith equation. We call the set of all a ij s arranged in a rectangular array the coefficient matrix of the system. Using matrix notation we can write the system as Ax b a 11 a 12 a 1n x 1 a 21 a 22 a 2n a 31 a 32 a 3n x a m1 a m2 a mn x n b 1 b 2.. We define the augmented coefficient matrix for the system as b m (2) Â 1.2. Row-echelon form of a matrix. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b a m1 a m2 a mn b m (3) Leading zeroes in the row of a matrix. A row of a matrix is said to have k leading zeroes if the first k elements of the row are all zeroes and the (k+1)th element of the row is not zero Row echelon form of a matrix. A matrix is in row echelon form if each row has more leading zeroes than the row preceding it. Date: August 20,

2 2 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Examples of row echelon matrices. The following matrices are all in row echelon form A B C (4) Pivots. The first non-zero element in each row of a matrix in row-echelon form is called a pivot. For the matrix A above the pivots are 3,5,4. For the matrix B they are 1,2 and for C they are 1,4,3. For the matrices B and C there is no pivot in the last row Reduced row echelon form. A row echelon matrix in which each pivot is a 1 and in which each column containing a pivot contains no other nonzero entries, is said to be in reduced row echelon form. This implies that columns containing pivots are columns of an identity matrix. The matrices D and E below are in reduced row echelon form. D (5) E The matrix F is in row echelon form but not reduced row echelon form F (6) Rank. The number of non-zero rows in the row echelon form of a matrix A produced by elementary operations on A is called the rank of A. Matrix D in equation (5) has rank 3, matrix E has rank 2, while matrix F in (6) has rank Solutions to equations (stated without proof). a: A system of linear equations with coefficient matrix A, right hand side vector b, and augmented matrix Â has a solution if and only if rank (A) rank (Â)

3 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 3 b: A linear system of equations must have either no solution, one solution, or infinitely many solutions. c: If a linear system has exactly one solution, then the coefficient matrix A has at least as many rows as columns. A system with a unique solution must have at least as many equations as unknowns. d: If a system of linear equations has more unknowns than equations, it must either have no solution or infinitely many solutions. e: A coefficient matrix is nonsingular, that is, the corresponding linear system has one and only one solution for every choice of right hand side b 1,b 2,...,b m, if and only if number of rows of A number of columns of A rank (A) 1.3. Systems of linear equations and determinants Solving simple 2x2 systems using elementary row operations. Consider the following simple 2x2 system of linear equations We can write this in matrix form as a 11 x 1 + a 12 x 2 b 1 (7) a 21 x 1 + a 22 x 2 b 2 Ax b [ a11 a A 12,x a 21 a 22 [ x1,b x 2 [ b1. (8) b 2 If we append the column vector b to the matrix A, we obtain the augmented matrix for the system. This is written as Ã [ a11 a 12 b 1 a 21 a 22 b 2 (9) We can perform row operations on this matrix to reduce it to reduced row echelon form. We will do this in steps. The first step is to divide each element of the first row by a 11. This will give Ã 1 [ 1 a 12 b 1 a 11 a 11 a 21 a 22 b 2 (10) Now multiply the first row by a 21 to yield and subtract it from the second row [ a 21 a 21 a 12 a 11 a 21 b 1 a 11

4 4 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS a 21 a 22 b 2 a 21 a 12 a 21 b 1 a 21 a 11 a 11 0 a 22 a 21 a 12 b 2 a 21 b 1 a 11 a 11 This will give a new matrix on which to operate. [ a 1 12 b 1 a Ã 2 11 a 11 a 0 11 a 22 a 21 a 12 Now multiply the second row by Now multiply the second row by a12 a 11 to yield. row by a12 a 11 a 11 a 11 b 2 a 21 b 1 a 11 a 11 to obtain [ a 1 12 b 1 Ã 3 a 11 a 11 a b 2 a 21 b 1 (11) and subtract it from the first row. First multiply the second [ 0 a 12 a 12 ( a 11 b 2 a 21 b 1 ) a 11 a 11 ( ) Now subtract the expression in equation 12 from the first row of Ã3 to obtain the following row. [ 1 [ a 12 b 1 a 11 a 11 0 [ a 12 a 12 (a 11 b 2 a 21 b 1) a 11 a 11 (a 11 a 22 a 21 a 12) 1 0 b 1 a 11 a12 (a11 b2 a21 b1) a 11 (a 11 a 22 a 21 a 12) Now replace the first row in Ã 3 with the expression in equation 13 of obtain Ã 4 [ b a12 ( a11 b2 a21 b1 ) Ã 4 a 11 a 11 ( ) 0 1 a 11 b 2 a 21 b 1 This can be simplified as by putting the upper right hand term over a common denominator, and canceling like terms as follows 1 0 Ã b 1 a 2 11 a22 b1 a11 a21 a12 a12 a2 11 b2 + a11 a12 a21 b1 a 2 11 ( a11 a22 a21 a12 ) a 11 b 2 a 21 b 1 (12) (13) (14) b 1 a 2 11 a22 a12 a2 11 b2 a 2 11 ( a11 a22 a21 a12 ) a 11 b 2 a 21 b 1 (15) b 1 a 22 a 12 b 2 a 11 b 2 a 21 b 1 We can now read off the solutions for x 1 and x 2. They are

5 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 5 x 1 b 1 a 22 a 12 b 2 (16) x 2 a 11 b 2 a 21 b 1 Each of the fractions has a common denominator. It is called the determinant of the matrix A. The determinant of a 2x2 matrix A is given by det ( A ) A (17) If we look closely we can see that we can write the numerator of each expression as a determinant also. In particular x 1 b 1 a 22 a 12 b 2 x 2 a 11 b 2 a 21 b 1 b 1 a 12 b 2 a 22 A a 11 b 1 a 21 b 2 A The matrix of which we compute the determinant in the numerator of the first expression is the matrix A, where the first column has been replaced by the b vector. The matrix of which we compute the determinant in the numerator of the second expression is the matrix A where the second column has been replaced by the b vector. This procedure for solving systems of equations is called Cramer s rule and will be discussed in more detail later An example problem with Cramer s rule. Consider the system of equations (18) Using Cramer s rule 3x 1 +5x 2 11 (19) 8x 1 3x 2 13 b 1 a 12 b 2 a 22 x 1 A a 11 b 1 a 21 b 2 x 2 A ( 33) (65) ( 9) (40) (39) (88) ( 9) (40) (20) 2.1. Definition and analytical computation. 2. DETERMINANTS

6 6 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Definition of a determinant. The determinant of an nxn matrix A a ij, written A, is defined to be the number computed from the following sum where each element of the sum is the product of n elements: A (±) a 1i a 2 j...a nr, (21) the sum being taken over all n! permutations of the second subscripts. A term is assigned a plus sign if (i,j,...,r) is an even permutation of (1,2,...,n) and a minus sign if it is an odd permutation. An even permutation is defined as making an even number of switches of the indices in (1,2,...,n), similarly for an odd permutation. Thus each term in the summation is the product of n terms, one term from each column of the matrix Example 1. As a first example consider the following 2x2 case ( ) 1 2 A 2 5 Here a 11 1,a 12 2,a 21 2, and a There are two permutations of (1,2). 1: (1,2) is even (0 switches): + a 11 a 22 2: (2,1) is odd (1 switches): + a 12 a 21 Notice that the permutations are over the second subscripts. The determinant of A is then A +a 11 a 22 a 12 a (23) The 2 x 2 case is easy to remember since there are just 2 rows and 2 columns. Each term in the summation has two elements and there are two terms. The first term is the product of the diagonal elements and the second term is the product of the off-diagonal elements. The first term has a plus sign because the second subscripts are 1 and 2 while the second term has a negative sign because the second subscripts are 2 and 1 (one permutation of 1 and 2) Example 2. Now consider another 2x2 example ( ) 3 5 A 8 3 Here a 11 3,a 12 5,a 21 8, and a The determinant of a is given by A +a 11 a 22 a 12 a 21 (3) ( 3) (8)(5) 49 (25) Example 3. Consider the following 3x3 case A (26) The indices are the numbers 1, 2 and 3. There are 6 (3!) permutations of (1, 2, 3). They are as follows: (1, 2, 3), (2, 1, 3), (1, 3, 2), (3, 2, 1), (2, 3, 1) and (3, 1, 2). The three in bold come from switching one pair of indices in the natural order (1, 2, 3). The final two come from making a switch in one of the bold sets. Specifically 1: (1, 2, 3) is even (0 switches): + a 11 a 22 a 33 2: (2, 1, 3) is odd (1 switches): - a 12 a 21 a 33 3: (1, 3, 2) is odd (1 switches): - a 11 a 23 a 32 (22) (24)

7 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 7 4: (3, 2, 1) is odd (1 switches): - a 13 a 22 a 31 5: (2, 3, 1) is even (2 switches): + a 12 a 23 a 31 6: (3, 1, 2) is even (2 switches): + a 13 a 21 a 32 Notice that the permutations are over the second subscripts, that is, every term is of the form a 1. a 2. a 3. with the second subscript varying. The determinant of A is then A + a 11 a 22 a 33 a 12 a 21 a 33 a 11 a 23,a 32 a 13 a 22 a 31 (27) +a 12 a 23 a 31 +a 13 a 21 a The first subscripts in each term do not change but the second subscripts range over all permutations of the numbers 1, 2, 3,..., n. The first term is the product of the diagonal elements and has a plus sign because the second subscripts are (1, 2, 3). The last term also has a plus sign because the second subscripts are (3, 1, 2). This permutation come from 2 switches [(1, 2, 3) (3,2,1) (3,1,2). A simple way to remember this formula for a 3x3 matrix is to use diagram in figure 1. The elements labeled 1j are multiplied together and receive a plus sign in the summation. FIGURE 1. 3 x 3 Determinant Plus Signs

8 8 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS The elements labeled 2j are multiplied together and receive a minus sign in the summation as in figure 2. FIGURE 2. 3 x 3 Determinant Minus Signs Example 4. Consider the following 4x4 case a 11 a 12 a 13 a 14 A a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 (28) a 41 a 42 a 43 a 44 The indices are the numbers 1, 2 3, and 4. There are 24 (4!) permutations of (1, 2, 3,4). They are contained in table 1. Notice that the permutations are over the second subscripts, that is, every term is of the form a 1. a 2. a 3. a 4. with the second subscript varying. The general formula is rather difficult to remember or implement for large n and so another method is normally used Submatrices and partitions Submatrix. A submatrix is a matrix formed from a matrix A by taking a subset consisting of j rows with column elements from a set k of the columns. For example consider A({1,3},{2,3}) below ( ) A A({1, 3}, {2, 3}) (29)

9 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 9 TABLE 1. Determinant of a 4x4 Matrix Permutation Sign element of sum Permutation Sign element of sum a 11 a 22 a 33 a a 13 a 21 a 34 a a 11 a 22 a 34 a a 13 a 21 a 32 a a 11 a 23 a 32 a a 13 a 22 a 31 a a 11 a 23 a 34 a a 13 a 22 a 34 a a 11 a 24 a 33 a a 13 a 24 a 32 a a 11 a 24 a 32 a a 13 a 24 a 31 a a 12 a 21 a 33 a a 14 a 21 a 32 a a 12 a 21 a 34 a a 14 a 21 a 33 a a 12 a 23 a 34 a a 14 a 22 a 33 a a 12 a 23 a 31 a a 14 a 22 a 31 a a 12 a 24 a 31 a a 14 a 23 a 31 a a 12 a 24 a 33 a a 14 a 23 a 32 a 41 The notation A({1,3},{2,3}) means that we take the first and third rows of A and include the second and third elements of each row Principal submatrix. A principal submatrix is a matrix formed from a square matrix A by taking a subset consisting of n rows and column elements from the same numbered columns. For example consider A({1,3},{1,3}) below A ) A({1, 3}, {1, 3})( (30) Minor. A minor is the determinant of a square submatrix of the matrix A. For example consider A({2, 3}, {1, 3}) A A({2, 3}, {1, 3}) ( ) 2 2 A({2, 3}, {1, 3}) 6 (31) Principal minor. A principal minor is the determinant of a principal submatrix of A. For example consider A({1, 2}, {1, 2}). A ) A({1, 2}, {1, 2})( A({1, 2}, {1, 2}) 7 (32) Cofactor. The cofactor (denoted A ij ) of the element a ij of any square matrix A is (-1) i+j times the minor of A that is obtained by including all but the ith row and the jth column, or alternatively the minor that is obtained by deleting the ith and jth rows. For example the cofactor of a 12 below is found as

10 10 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS a 11 a 12 a A a 21 a 22 a (33) a 31 a 32 a ( ) 2 2 A({2, 3}, {1, 3} 1 4 A 12 ( 1) 3 A({2, 3}, {1, 3} ( 1) ( 1) 3 (6) Computing determinants using cofactors Definition of a cofactor expansion. The determinant of a square matrix A can be found inductively using the following formula n det A A a ij A ij (34) where j denotes the jth column of the matrix a. This is called an expansion of A by column j of A. The result is the same for any other column. This can also be done for rows letting the sum range over j instead of i Examples. 1: Consider as an example the following 2x2 matrix and expand using the first row ( ) 4 3 A 1 2 i1 A 4 ( 1) 2 (2) + 3 ( 1) 3 (1) where the cofactor of 4 is (-1) (1+1) times the submatrix that remains when we delete row and column 1 from the matrix A while the cofactor of 3 is (-1) (1+2) times the submatrix that remains when we delete row 1 and column 2 from the matrix A. In this case each the principle submatrices are just single numbers so there is no need to formally compute a determinant. 2: Now consider a 3x3 example computed using the first row of the matrix. A A 1 ( 1) ( 1) ( 1)4 1 0 (1) (20) + ( 2) ( 2) + 3 ( 5) 9 We can also compute it using the third row of the matrix

11 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 11 A A 1 ( 1) ( 1) ( 1)6 0 5 (1) ( 11) + (0) (2) + (4) (5) 9 or the first column A A 1 ( 1) ( 1) ( 1)4 5 2 (1) (20) + (0) ( 8) + (1) ( 11) Expansion by alien cofactors Definition of expansion by alien cofactors. When we expand a row of the matrix using the cofactors of a different row we call it expansion by alien cofactors. a kj A ij 0k i (35) j Some examples. 1: example Expand row 1 using the 2nd row cofactors to obtain (4) (-1) (2+1) (3) + (3) (-1) (2+2 (4) : example

12 12 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Expand row 1 using the 2nd row cofactors. The second row cofactors are ( 1) (2+1) and ( 1) (2+2) and ( 1) (2+3) This then gives for the cofactor expansion(4) (-1) (0) + (3) (1) (3) + (1) (-1) (9) General principle. If we expand the rows of a matrix by alien cofactors, the expansion will equal zero Singular and nonsingular matrices. The square matrix A is said to be singular if A 0, and nonsingular if A Determinants, Minors, and Rank. Theorem 1. The rank of an m n matrix A is k if and only if every minor in A of order k +1vanishes, while there is at least one minor of order k which does not vanish. Proposition 1. Consider an m n matrix A. 1: det A 0 if every minor of order n - 1 vanishes. 2: If every minor of order n equals zero, then the same holds for the minors of higher order. 3 (restatement of theorem): The largest among the orders of the non-zero minors generated by a matrix is the rank of the matrix Basic rules for determinants. 1: If a row or column of A is all zeroes then A 0. 2: The determinant of A is equal to the determinant of A, i.e., A A. 3: If B is the matrix obtained by multiplying one row or column of A by the same number α, then B α A. 4: If two rows or columns of A are interchanged, then the determinant of A changes sign, but keeps its absolute value. 5: If two rows or columns of A are equal, then A 0. 6: If two rows or columns of A are proportional, then A 0. 7: 7. If a scalar multiple of one row ( or column) of A is added to another row (or column) of A, then the value of the determinant does not change. 8: If A and B are both nxn then AB A B. 9: If A is an nxn matrix and α is a real number then αa α n A. 10: The determinant of a sum is not necessarily the sum of the determinants. 11: The determinant of a diagonal matrix is the product of the diagonal elements. 12: The determinant of an upper or lower triangular matrix is the product of the diagonal elements Some problems to solve. Provide an example for each of items 1-12 in section 2.6.

13 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS THE INVERSE OF A MATRIX A 3.1. Definition of the inverse of a matrix. Given a matrix A, if there exists a matrix B such that AB BA I (36) then we say that B is the inverse of A. Moreover, A is said to be invertible in this case. Because BA AB I, the matrix A is also an inverse of B - that is, A and B are inverses of each other. Inverses are only defined for square matrices. We usually denote the inverse of A by A 1 and write A 1 A AA 1 I (37) 3.2. Some examples of matrix inversion. Show that the following pairs of matrices are inverses of each other. [ [ Existence of the inverse. A square matrix A has an inverse A 0. A square matrix is said to be singular if A 0and nonsingular if A 0. Thus a matrix has an inverse if and only if it is nonsingular Uniqueness of the inverse. The inverse of a square matrix is unique if it exists Some implications of inverse matrices. AX I X A 1 (38) BA I B A Properties of the inverse. Let A and B be invertible nxn matrices. 1: A 1 is invertible and (A 1 ) 1 A 2: AB is invertible and (AB) 1 B 1 A 1 3: A is invertible and (A 1 ) (A ) 1 4: (ABC) 1 C 1 B 1 A 1, if A, B, and C have inverses 5: (ca) 1 c 1 A 1 whenever c is a number 0 6: In general, (A + B) 1 A 1 +B 1 7: (A 1 ) (A ) Orthogonal matrices. A matrix A is called orthogonal if its inverse is equal to its transpose, that is if A 1 A (39)

14 14 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 3.8. Finding the inverse of a 2x2 matrix. For a general 2x2 matrix A, we can find the inverse using the following matrix equation AB I (40) [ [ [ a11 a 12 b11 b a 21 a 22 b 21 b We can divide this into two parts as follows [ [ a11 a 12 b11 a 21 a 22 b 21 [ a11 a 12 a 21 [ b12 a 22 b 22 [ 1 0 and [ 0 1 (41) We can solve each system using Cramer s rule. For the first system we obtain 1 a 12 0 a 22 b 11 a 11 a 12 a 21 a 22 and a 11 1 a 21 0 b 21 a 11 a 12 a 21 a 22 In a similar fashion we can show that a 22 (42) a 21 b 12 a 12 (43) Combining the expressions we see that a 11 b 22 [ A 1 1 a22 a 12 (44) a 11 a 22 a 12 a 21 a 21 a 11 This leads to the familiar rule that to compute the determinant of a 2x2 matrix we switch the diagonal elements, change the sign of the off-diagonal ones and divide by the determinant. 4. SOLVING EQUATIONS BY MATRIX INVERSION 4.1. Procedure for solving equations using a matrix inverse. Let A be an nxn matrix. Let B be an arbitrary matrix. Then we ask whether there are matrices C and D of suitable dimension such that

15 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 15 AC B (45) DA B In the first case the matrix B must have n rows, while in the second B must have n columns. Then we have the following theorem. Theorem 2. If A 0, then: AC B C A 1 B (46) DA B D BA 1 As an example, solve the following system of equations using theorem 2. 3 x 1 +5x 2 14 (47) 8 x 1 2 x 2 22 We can write the system as Ax b [ 3 5 A,x 8 2 x A 1 b A 1 [ [ x1 We can compute A 1 from equation 44. A 1 1 (3) ( 2) (8) (5) 1 [ [ We can now compute A 1 b as follows A 1 b [ [ ,b x 2 [ [ (48) (49) [ [ [ (50) Some example systems. Solve each of the following using matrix inversion 3x 1 + x 2 6 5x 1 x 2 2 4x 1 + x x 1 +2x x 1 +3x 2 10 x 1 +2x 2 5 (51) (52) (53)

16 16 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 4x 1 +3x 2 11 x 1 +2x 2 4 (54) 5. A GENERAL FORMULA FOR THE INVERSE OF A MATRIX USING ADJOINT MATRICES 5.1. The adjoint of a matrix. The adjoint of the matrix A denoted adj (A) or A + is the transpose of the matrix obtained from A by replacing each element a ij by its cofactor A ij. For example consider the matrix A (55) Now find the cofactor of each element. For example the cofactor of the 1 in the upper left hand corner is computed as A 11 ( 1) (1+1) (56) Similarly the cofactor of a 23 2 is given by The entire matrix of cofactors is given by A 23 ( 1) (2+3) A 11 A A 1n A 21 A A 2n.. A n1 A n2... A nn We can then compute the adjoint by taking the transpose 2 (57) (58) A + A 11 A A 1n A 21 A A 2n.. A n1 A n2... A nn (59) 5.2. Finding an inverse matrix using adjoints. For a square nonsingular matrix A, its inverse is given by A 1 1 A A+ (60)

17 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS An example of finding the inverse using the adjoint. First find the cofactor matrix. A A ij (61) Now find the transpose of A ij A (62) The determinant of A is A 9 Putting it all together we obtain A /9 8/9 11/ /9 1/9 2/ /9 2/9 5/9 (63) We can check the answer as follows /9 8/9 11/9 9/9 0 0 AA /9 1/9 2/9 0 9/9 0 I (64) /9 2/9 5/ /9 6. FINDING THE INVERSE OF A MATRIX USING ELEMENTARY ROW OPERATIONS The most effective computational way to find the inverse of a matrix is to use elementary row operations. This is done by forming the augmented matrix [A : I similar to the augmented matrix we use when solving a system equation, except that we now add n columns of the identity matrix instead of the right hand side vector in the equations system. In effect we are writing the system AX I (65) and then solving for the matrix X, which will be A 1. Consider the following matrix A (66) Now append an identity matrix as follows Ã (67) Subtract the first row from the last row to obtain Ã (68) Now divide the second row by 4

18 18 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Ã /4 0 1/4 0 (69) Now multiply the second row by 2 and add to the third row Ã /4 0 1/4 0 (70) 0 0 1/2 1 1/2 1 Now multiply the third row by 2 Ã /4 0 1/4 0 (71) Now multiply the second row by 2 and subtract from the first 1 0 3/2 1 1/2 0 Ã /4 0 1/4 0 (72) Now multiply the third row by 3/2 and subtract from the first Ã /4 0 1/4 0 (73) Finally multiply the third row by 1/4 and subtract from the second Ã /2 0 1/2 (74) The inverse of A is then given by A 1 1/2 0 1/2 (75) Consider the general equation system 7. THE GENERAL FORM OF CRAMER S RULE where A is nxn, x in nx1, and b is nx1. Ax b (76) Let D j denote the determinant formed from A by replacing the jth column with the column vector b. Thus a 11 a a 1 j 1 b 1 a 1 j+1 a 1n D j a 21 a a 2 j 1 b 2 a 2 j+1 a 2n (77) a n1 a n2... a nj 1 b n a nj+1 a nn Then the general system of n equations in n unknowns has unique solution if A 0. The solution is

19 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 19 x 1 D 1 A,x 2 D 2 A,,x n D n A (78) 8. A NOTE ON HOMOGENEOUS SYSTEMS OF EQUATIONS The homogeneous system of n equations in n unknowns or Ax 0 (79) a 11 x 1 + a 12 x a 1n x n b 1 a 21 x 1 + a 22 x a 2n x n b 2 a 31 x 1 + a 32 x a 3n x n b (80) a m1 x 1 + a m2 x a mn x n b m has a nontrivial solution (a trivial solution has x 1 0,x 2 0,... x n 0 if and only if A 0.

20 20 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Lecture Questions 1: Provide two examples of linear systems that have no solution and explain why. 2: Provide two examples of linear systems that have exactly one solution and explain why. 3: Provide two examples of linear systems that have exactly infinitely many solutions and explain why. Provide at least one of these where the number of equations and unknowns is the same. 4: Explain why if a linear system has exactly one solution, the coefficient matrix A must have at least as many rows as columns. Give examples of why this is the case. 5: If a system of linear equations has more unknowns than equations, it must either have no solution or infinitely many solutions. Explain why. Provide at least two examples.

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