If A = bmatrix 2 & 3 5 & -2 bmatrix , B⁻¹ = bmatrix 1 5 & 2 5 2 5 & - 1 5 bmatrix , then (AB)⁻¹ =

Step-by-step solution

Using the reversal law for the inverse of a product of matrices, we have: (AB)⁻¹ = B⁻¹A⁻¹ First, we find the inverse of matrix A . The determinant of A is: |A| = (2)(-2) - (3)(5) = -4 - 15 = -19 The adjugate of A is obtained by swapping the diagonal elements and changing the signs of the off-diagonal elements: adj (A) = bmatrix -2 & -3 -5 & 2 bmatrix Thus, A⁻¹ is given by: A⁻¹ = 1 |A| adj (A) = 1 -19 bmatrix -2 & -3 -5 & 2 bmatrix = 1 19 bmatrix 2 & 3 5 & -2 bmatrix We are given B⁻¹ = bmatrix 1 5 & 2 5 2 5 & - 1 5