If A= [ array ccc 4 & & -3 0 & 2 & 5 1 & 1 & 3 array ] then A⁻¹ exists if

Step-by-step solution

A matrix A is invertible if and only if its determinant is non-zero, i.e., |A| 0 . The determinant of matrix A = bmatrix 4 & & -3 0 & 2 & 5 1 & 1 & 3 bmatrix is calculated by expanding along the first column: |A| = 4(2 3 - 5 1) - 0( 3 - (-3) 1) + 1( 5 - (-3) 2) |A| = 4(6 - 5) - 0 + 1(5 + 6) |A| = 4(1) + 5 + 6 = 5 + 10 For A⁻¹ to exist, |A| 0 . 5 + 10 0 5 -10 -2 Answer: -2