If A= [ array lll 5 & 0 & 4 2 & 3 & 2 1 & 2 & 1 array ] B⁻¹= [ array lll 1 & 3 & 3 1 & 4 & 3 1 & 3 & 4 array ] then (A B)⁻¹ is equal to

Step-by-step solution

We know that (AB)⁻¹ = B⁻¹ A⁻¹ . First, calculate A⁻¹ . The determinant of A is |A| = 5(3-4) - 0(2-2) + 4(4-3) = 5(-1) + 4(1) = -5 + 4 = -1 . The cofactor matrix of A is C = bmatrix (3-4) & -(2-2) & (4-3) -(0-8) & (5-4) & -(10-0) (0-12) & -(10-8) & (15-0) bmatrix = bmatrix -1 & 0 & 1 8 & 1 & -10 -12 & -2 & 15 bmatrix . The adjoint matrix adj(A) = C^T = bmatrix -1 & 8 & -12 0 & 1 & -2 1 & -10 & 15 bmatrix . A⁻¹ = 1 |A| adj(A) = -1 bmatrix -1 & 8 & -12 0 & 1 & -2 1 & -10 & 15 bmatrix = bmatrix 1 & -8 & 12 0 & -1 & 2 -