Answer:
So we have shown that (AB)^-1 = B^-1 A^-1.
Explanation:
Okay, let's solve this step-by-step:
1) We are given the matrices:
A = [[1, 2], [-3, 4]]
B = [[4, 3], [-2, 1]]
2) To find (AB)^-1, we first need to find AB. Matrix multiplication gives:
AB = [[1*4 + 2*-2, 1*3 + 2*1], [-3*4 + 4*-2, -3*3 + 4*1]]
= [[4, 7], [-12, 1]]
3) Next, we need to find the inverses of A and B:
A^-1 = [[4, -2], [-3, 1]] / (4*4 - 4*(-3)) = [[4, -2], [-3, 1]] / 16 = [[ 0.25, -0.125], [-0.1875, 0.0625]]
B^-1 = [[1, -2], [3, -1]] / (1*1 - 1*(3)) = [[1, -2], [3, -1]] / -2 = [[-0.5, 1], [-1.5, 0.5]]
4) Finally, we can multiply the inverses:
(AB)^-1 = A^-1 * B^-1
= [[ 0.25, -0.125], [-0.1875, 0.0625]] * [[-0.5, 1], [-1.5, 0.5]]
= [[0.125, -0.25], [0.28125, -0.03125]]
Therefore, (AB)^-1 = [[0.125, -0.25], [0.28125, -0.03125]]
So we have shown that (AB)^-1 = B^-1 A^-1.