911 9.3 Determinant Solution of Linear Systems Determinant Theorems The following theorems hold true for square matrices of any dimension and can be used to simplify finding determinants. Determinant Theorems 1. If every element in a row (or column) of matrix A is 0, then A = 0. 2. If the rows of matrix A are the corresponding columns of matrix B, then B = A . 3. If any two rows (or columns) of matrix A are interchanged to form matrix B, then B = - A . 4. Suppose matrix B is formed by multiplying every element of a row (or column) of matrix A by the real number k. Then B = k # A . 5. If two rows (or columns) of matrix A are identical, then A = 0. 6. Changing a row (or column) of a matrix by adding to it a constant times another row (or column) does not change the determinant of the matrix. 7. If matrix A is in triangular form, having only zeros either above or below the main diagonal, then A is the product of the elements on the main diagonal of A. EXAMPLE 4 Using the Determinant Theorems Use the determinant theorems to evaluate each determinant. (a) 3 -2 4 2 6 7 3 0 16 83 (b) 4 3 -7 4 10 0 1 8 3 0 0 -5 2 0 0 0 64 SOLUTION (a) Use determinant theorem 6 to obtain a 0 in the second row of the first column. Multiply each element in the first row by 3, and add the result to the corresponding element in the second row. 3 -2 4 2 0 19 9 0 16 83 3R1 + R2 Now, find the determinant by expanding by column 1. -21-121+1 2 19 9 16 82 = -2112182 = -16 19182 - 16192 = 152 - 144 = 8 (b) Use determinant theorem 7 to find the determinant of this triangular matrix by multiplying the elements on the main diagonal. 4 3 -7 4 10 0 1 8 3 0 0 -5 2 0 0 0 6 4 = 31121-52162 = -90 S Now Try Exercises 51 and 53.
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