Let A = pmatrix 2 & -2 1 & -1 pmatrix and B = pmatrix -1 & 2 -1 & 2 pmatrix . If (2A + 4B)⁻³ = a 64 A + b 64 B , where a and b are integers, then the value of a + b is _____.

Step-by-step solution

Given matrices are A = pmatrix 2 & -2 1 & -1 pmatrix and B = pmatrix -1 & 2 -1 & 2 pmatrix . First, we check the properties of A and B : A^2 = pmatrix 2 & -2 1 & -1 pmatrix pmatrix 2 & -2 1 & -1 pmatrix = pmatrix 4-2 & -4+2 2-1 & -2+1 pmatrix = pmatrix 2 & -2 1 & -1 pmatrix = A B^2 = pmatrix -1 & 2 -1 & 2 pmatrix pmatrix -1 & 2 -1 & 2 pmatrix = pmatrix 1-2 & -2+4 1-2 & -2+4 pmatrix = pmatrix -1 & 2 -1 & 2 pmatrix = B Also, A + B = pmatrix 1 & 0 0 & 1 pmatrix = I . Since A and B are idempotent and sum to the identit