APPENDIX F: MATRICES, DETERMINANTS, AND SYSTEMS OF EQUATIONS

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1 APPENDIX F: MATRICES, DETERMINANTS, AND SYSTEMS OF EQUATIONS F1 MATRIX DEFINITIONS AND NOTATIONS MATRIX An m n matrix is a rectangular or square array of elements with m rows and n columns An example of a matrix is shown in Eq (F1) a 11 a 1 a 1n a 1 a a n A ¼ 7 (F1) 5 a m1 a m a mn For each subscript, a ij ; i ¼ the row, and j ¼ the column If m ¼ n, the matrix is said to be a square matrix VECTOR If a matrix has just one row, it is called a row vector An example of a row vector follows: B ¼ ½b 11 b 1 b 1n Š (F) If a matrix has just one column, it is called a column vector An example of a column vector follows: c 11 c 1 C ¼ 7 (F) 5 c m1 1

2 Appendix F: Matrices, Determinants, and Systems of Equations PARTITIONED MATRIX A matrix can be partitioned into component matrices or vectors For example, let a 11 a 1 a 1 a 1 a 11 a 1 a 1 a 1 a A ¼ 1 a a a 7 a 1 a a a 5 ¼ a 1 a a a 7 a 1 a a a 5 ¼ A 11 A 1 (F) A 1 A a 1 a a a a 1 a a a where a 11 a 1 a 1 a 1 A 11 ¼ a 1 a 5; A 1 ¼ a a 5 a 1 a a a A 1 ¼ ½a 1 a Š; A ¼ ½a a Š NULL MATRIX A matrix with all elements equal to zero is called the null matrix; that is, a ij ¼ 0 for all i and j An example of a null matrix follows: A ¼ 7 5 (F5) DIAGONAL MATRIX A square matrix with all elements off of the diagonal equal to zero is said to be a diagonal matrix; that is, a ij ¼ 0 for i ¼ j An example of a diagonal matrix follows: a a 0 0 A ¼ 0 0 a a nn 7 5 (F) IDENTITY MATRIX A diagonal matrix with all diagonal elements equal to unity is called an identity matrix and is denoted by I; that is, a ij ¼ 1 for i ¼ j, and a ij ¼ 0 for i ¼ j An example of an identity matrix follows: A ¼ 7 5 (F7)

3 SYMMETRIC MATRIX A square matrix for which a ij ¼ a ji is called a symmetric matrix An example of a symmetric matrix follows: 8 7 A ¼ (F8) 7 MATRIX TRANSPOSE The transpose of matrix A, designated A T, is formed by interchanging the rows and columns of A Thus, if A is an m n matrix with elements a ij, the transpose is an n m matrix with elements a ji An example follows Given then A ¼ F1 Matrix Definitions and Notations (F9) A T ¼ (F10) 9 5 DETERMINANT OF A SQUARE MATRIX The determinant of a square matrix is denoted by det A, or a 11 a 1 a 1n a 1 a a n 7 5 a m1 a m1 a mn The determinant of a matrix, A ¼ a 11 a 1 a 1 a is evaluated as det A ¼ a 11 a 1 a 1 a ¼ a 11a a 1 a 1 (F11) (F1) (F1) MINOR OF AN ELEMENT The minor, M ij of element a ij of det A is the determinant formed by removing the ith row and the jth column from det A As an example, consider the following determinant: 8 7 det A ¼ 9 (F1) 5 1

4 Appendix F: Matrices, Determinants, and Systems of Equations The minor M is the determinant formed by removing the third row and the second column from det A Thus, M ¼ 7 ¼ (F15) COFACTOR OF AN ELEMENT The cofactor, C ij, of element a ij of det A is defined to be C ij ¼ð 1Þ ðiþ jþ M ij For example, given the determinant of Eq (F1) C 1 ¼ð 1Þ ðþ1þ M 1 ¼ð 1Þ ¼ 5 (F1) (F17) EVALUATING THE DETERMINANT OF A SQUARE MATRIX The determinant of a square matrix can be evaluated by expanding minors along any row or column Expanding along any row, we find det A ¼ å n a ik C ik (F18) k¼1 where n ¼ number of columns of A; j is the jth row selected to expand by minors; and C ik is the cofactor of a ik Expanding along any column, we find det A ¼ å m a kj C kj (F19) k¼1 where m ¼ number of rows of A; j is the jth column selected to expand by minors; and C kj is the cofactor of a kj For example, if 1 A ¼ (F0) 8 5 then, expanding by minors on the third column, we find det A ¼ ð 7Þ þ 1 5 ¼ 195 (F1) Expanding by minors on the second row, we find det A ¼ ð 5Þ 5 þ 1 8 ð 7Þ ¼ 195 (F) SINGULAR MATRIX A matrix is singular if its determinant equals zero NONSINGULAR MATRIX A matrix is nonsingular if its determinant does not equal zero

5 ADJOINT OF A MATRIX The adjoint of a square matrix, A, written adj A, is the matrix formed from the transpose of the matrix A after all elements have been replaced by their cofactors Thus, T C 11 C 1 C 1n C 1 C C n adj A ¼ 7 (F) 5 C n1 C n C nn For example, consider the following matrix: 1 A ¼ 1 55 (F) 8 7 Hence, T 7 8 adj A ¼ ¼ (F5) F Matrix Operations 5 RANK OF A MATRIX The rank of a matrix, A, equals the number of linearly independent rows or columns The rank can be found by finding the highest-order square submatrix that is nonsingular For example, consider the following: 1 5 A ¼ (F) 15 The determinant of A ¼ 0 Since the determinant is zero, the matrix is singular Choosing the submatrix A ¼ 1 5 (F7) 7 whose determinant equals 7, we conclude that A is of rank F MATRIX OPERATIONS ADDITION The sum of two matrices, written A þ B ¼ C,isdefinedbya ij þ b ij ¼ c ij Forexample, þ ¼ (F8) 5 1 8

6 Appendix F: Matrices, Determinants, and Systems of Equations SUBTRACTION The difference between two matrices, written A B ¼ C, is defined by a ij b ij ¼ c ij For example, ¼ 5 (F9) 5 7 MULTIPLICATION The product of two matrices, written AB ¼ C, is defined by c ij ¼ å n a ik b kj For example, if k¼1 A ¼ a b 11 a 1 a 11 b 1 b 1 1 ; B ¼ b a 1 a a 1 b b 5 (F0) b 1 b b then C ¼ ða 11b 11 þ a 1 b 1 þ a 1 b 1 Þ ða 11 b 1 þ a 1 b þ a 1 b Þ ða 11 b 1 þ a 1 b þ a 1 b Þ ða 1 b 11 þ a b 1 þ a b 1 Þ ða 1 b 1 þ a b þ a b Þ ða 1 b 1 þ a b þ a b Þ (F1) Notice that muitiplicalion is defined only if the number of columns of A equals the number of rows of B MULTIPLICATION BY A CONSTANT A matrix can be multiplied by a constant by multiplying every element of the matrix by that constant For example, if A ¼ a 11 a 1 (F) a 1 a then ka ¼ ka 11 ka 1 (F) ka 1 ka INVERSE An n n square matrix, A, has an inverse, denoted by A 1, which is defined by AA 1 ¼ I (F) where I is an n n identity matrix The inverse of A is given by A 1 ¼ adj A (F5) det A For example, find the inverse of A in Eq (F) The adjoint was calculated in Eq (F5) The determinant of A is det A ¼ ð 1Þ 8 7 þ 5 ¼ (F)

7 F Matrix and Determinant Identities 7 Hence, A 1 0:5 0:9 0:059 ¼ ¼ 1:088 0: 0:5 5 (F7) 0:91 0:118 0:17 F MATRIX AND DETERMINANT IDENTITIES The following are identities that apply to matrices and determinants MATRIX IDENTITIES Commutative Law A þ B ¼ B þ A AB ¼ BA (F8) (F9) Associative Law A þðb þ CÞ ¼ðA þ BÞþC AðBCÞ ¼ðABÞC (F0) (F1) Transpose of Sum ða þ BÞ T ¼ A T þ B T (F) Transpose of Product ðabþ T ¼ B T A T (F) DETERMINANT IDENTITIES Multiplication of a Single Row or Single Column of a Matrix, A, by a Constant If a single row or single column of a matrix, A, is multiplied by a constant, k, forming the matrix, ~ A, then det ~ A ¼ k det A (F) Multiplication of All Elements of an n n Matrix, A, by a Constant detðkaþ ¼k n det A (F5) Transpose det A T ¼ det A (F)

8 8 Appendix F: Matrices, Determinants, and Systems of Equations Determinant of the Product of Square Matrices det AB ¼ det A det B det AB ¼ det BA (F7) (F8) F SYSTEMS OF EQUATIONS REPRESENTATION Assume the following system of n linear equations: a 11 x 1 þ a 1 x þ þa 1n ¼ b 1 a 1 x 1 þ a x þ þa n ¼ b a n1 x 1 þ a n x þ þa nn ¼ b n This system of equations can be represented in vector-matrix form as Ax ¼ B where a 11 a 1 a 1n b 1 x 1 a 1 a a n A ¼ 7 5 ; B ¼ b 7 5 ; x ¼ x 7 5 a n1 a n a nn b n x n For example, the following system of equations, 5x 1 þ 7x ¼ 8x 1 þ x ¼ 9 can be represented in vector-matrix form as Ax ¼ B, or 5 7 x1 ¼ 8 9 x (F9) (F50) (F51a) (F51b) (F5) SOLUTION VIA MATRIX INVERSE If A is nonsingular, we can premultiply Eq (F50) by A 1, yielding the solution x Thus, x ¼ A 1 B For example, premultiplying both sides of Eq (F5) by A 1, where 1 A :05 0:091 ¼ ¼ 8 0:105 0:058 we solve for x ¼ A 1 B as follows: x 1 0:05 0:091 0:987 ¼ ¼ x 0:105 0: :7 (F5) (F5) (F55)

9 SOLUTION VIA CRAMER S RULE Equation (F5) allows us to solve for all unknowns, x i, where i ¼ 1ton If we are interested in a single unknown, x k, then Cramer s rule can be used Given Eq (F50), Cramer s rule states that x k ¼ det A k (F5) det A where A k ; is a matrix formed by replacing the kth column of A by B For example, solve Eq (F5) Using Eq (F5) with A ¼ we find and x 1 ¼ x ¼ ; B ¼ 9 ¼ 75 ¼ 0:987 (F57) 7 ¼ 1 7 ¼ :7 (F58) Bibliography 9 BIBLIOGRAPHY Dorf, R C Matrix Algebra A Programmed Introduction Wiley, New York, 199 Kreyszig, E Advanced Engineering Mathematics th ed Wiley, New York, 1979 Wylie, C R, Jr Advanced Engineering Mathematics 5th ed McGraw-Hill, New York, 198

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