909 9.3 Determinant Solution of Linear Systems To find the determinant of a 3 * 3 or larger matrix, first choose any row or column. Then the minor of each element in that row or column must be multiplied by +1 or -1, depending on whether the sum of the row number and column number is even or odd. The product of a minor and the number +1 or -1 is a cofactor. Cofactor Let Mij be the minor for element aij in an n * n matrix. The cofactor of aij, written Aij, is defined as follows. Ai j = 1 −12 i+j # Mi j EXAMPLE 2 Finding Cofactors of Elements Find the cofactor of each of the following elements of the given matrix. £6 8 1 2 9 2 4 3 0§ (a) 6 (b) 3 (c) 8 SOLUTION (a) The element 6 is in the first row, first column of the matrix, so i = 1 and j = 1. M11 = 2 9 2 3 02 = -6. The cofactor is 1-121+11-62 = 11-62 = -6. (b) Here i = 2 and j = 3, so M23 = 2 6 1 2 22 = 10. The cofactor is 1-122+31102 = -11102 = -10. (c) We have i = 2 and j = 1, and M21 = 2 2 2 4 02 = -8. The cofactor is 1-122+11-82 = -11-82 = 8. S Now Try Exercise 17. Finding the Determinant of a Matrix Multiply each element in any row or column of the matrix by its cofactor. The sum of these products gives the value of the determinant. n : n Determinants The determinant of a 3 * 3 or larger matrix is found as follows. The process of forming this sum of products is called expansion by a given row or column.
RkJQdWJsaXNoZXIy NjM5ODQ=