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CHAPTER 11 Matrix Algebra. Networks
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A determinant of order n is a square array of elements having n rows and n columns, there thus being n elements in each row and each column. Now let aij denote the element at the intersection of the i th row and j th column. If we then strike out the row and column in which aij appears, the determinant that remains is of order n 1 and is called the minor determinant of element aij . In Chap. 3 we showed that the value of a determinant is equal to the sum of the products of the elements of any row or column and their corresponding minor determinants, each such product being multiplied by 1 i j . Thus, if we expand the thirdorder determinant in the last expression for x1 above, using minors of the rst column, we have that           a22 a23     y2  a12 a13  y3  a12 a13  1  x 1 y1    a a a32 a33 a23  D 32 a33 22 Likewise, if we expand the third-order determinant in the last expression for x2 , in terms of the minors of the second column, we have that           a21 a23     y2  a11 a13  y3  a11 a13  1 x2 y1    a a a a33 a23  D 31 31 a33 21 and lastly, if we expand the third-order determinant in the last expression for x3 , in terms of the minors of the third column, we have that           a21 a22     y2  a11 a12  y3  a11 a12  1 x 3 y1    a a a a32 a22  D 31 31 a32 21 Now, in the last three equations above, let us denote the value of each second-order minor determinant, including the sign factor 1 i j , by the notation Aij , where i and j are the numbers of the row and column struck out to form the minor determinant.* Using this notation, the last three equations above become A11 A A y 21 y2 31 y3 D 1 D D A A A x2 12 y1 22 y2 32 y3 D D D A A A x3 13 y1 23 y2 33 y3 D D D x1 which in matrix notation becomes 2 3 2 A11 x1 6 7 16 x2 5 4 A12 4 D x3 A13 32 3 479
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A21 A22 A23
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76 7 A32 54 y2 5 A33 y3
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Equation (479) above is the inverse form of eqs. (476) and (477); that is, (479) is of the form X A 1 Y 480
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* Aij is called the cofactor of element aij . If Mij is the minor determinant of any element aij , the cofactor of aij is the minor determinant with proper sign included; thus Aij 1 i j Mij
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where X and Y are the x and y column matrices. Comparison of eqs. (479) and (480) shows that A 1 , the INVERSE OF THE SQUARE MATRIX A of eqs. (476) and (477), is equal to 3 2 A11 A21 A31 16 7 A 1 4 A12 A22 A32 5 481 D A13 A23 A33 where D value of the determinant formed from the constant a coe cients of the original three simultaneous linear equations (eq. (475)), and where Aij value of the cofactor formed by deleting the row and column of element aij in the original determinant formed from the a coe cients. It should be understood that all subscripts, everywhere, refer to the i th row and j th column of the original simultaneous equations (475). Thus, in (481), A21 is actually the cofactor of a21 in the second row and rst column in the original eq. (475), even though A21 appears in the rst row, second column position in (481). The procedure will be clear from the discussion that follows eq. (482) below. It s apparent that the foregoing work can be extended to nding the inverse of any nth order square matrix A. Thus, given any square matrix A of order n, as in eq. (482) below, 3 2 a11 a12 a13 a1n 7 6 6 a21 a22 a23 a2n 7 7 6 7 6 482 A 6 a31 a32 a33 a3n 7 6 . . 7 . . 6 . . 7 . . . 5 . . 4 . an1 an2 an3 ann The procedure for nding the inverse matrix A 1 , necessary to satisfy the relationships AX Y X A 1 Y can be summarized in the following steps. Step 1 Find D, the value of the nth-order determinant formed from the elements of the given square matrix A of eq. (482). Step 2 Replace each element in the given matrix A by its cofactor to get a matrix which we ll call A0 ( A sub zero ); thus 3 2 A11 A12 A13 A1n 7 6A 6 21 A22 A23 A2n 7 6 . A0 6 . 7 . . . 7 . . 4 . . 5 . . . An1 An2 An3 Ann
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483
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Now interchange the rows and columns in (483) (that is, let the rst row become the rst column, the second row become the second column, and so on)* and then multiply by 1/D. The result is the inverse matrix A 1 of the given matrix A; that is, the matrix capable of transforming eq. (477) into (478).
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