The determinant of a 3matrix Mathematics （3矩阵的行列式数学）.pdf

The determinant of a 3 × 3 matrix sigma-matrices9-2009-1 We have seen that determinants are important in the solution of simultaneous equations and in finding inverses of matrices. The rule for evaluating the determinant of 2 × 2 matrices is quite straightforward (if rather unexpected). To evaluate the determinant of a 3 × 3 matrix is somewhat more complicated and relies on some other quantities known as minors and cofactors. Throughout this leaflet we will work with the 3 × 3 matrix  7 2 1  A =  0 3 −1   −3 4 −2  Minors Each element in a square matrix has its own minor. The minor is the value of the determinant of the matrix that results from crossing out the row and column of the element under consideration. For the element 7 in matrix A, since this element is in the first row and first column, we cross out the first row and column of A to leave the 2 × 2 matrix 3 −1 . We now evaluate the determinant of 4 −2 this matrix: 3 −1 = (3 × −2) − (−1 × 4) = −6 − (−4) = −2 4 −2 The minor of the element 7 is thus −2. For the element 4, since this element is in the third row and second column, we cross out the third row and second column of A to leave the 2 × 2 matrix 7 1 . We now evaluate the determinant 0 −1 of this matrix: 7 1 = (7 × −1) − (1 × 0) = −7 − (0) = −7 0 −1 The