Let A = pmatrix 4 & -12 1 & -3 pmatrix and B = pmatrix -3 & 12 -1 & 4 pmatrix . If M = _ k=1 ¹⁰ (A^k + k B^k) , then the sum of all elements of the matrix M is _____.

Step-by-step solution

Given matrices are A = pmatrix 4 & -12 1 & -3 pmatrix and B = pmatrix -3 & 12 -1 & 4 pmatrix . First, we find A^2 and B^2 : A^2 = pmatrix 4 & -12 1 & -3 pmatrix pmatrix 4 & -12 1 & -3 pmatrix = pmatrix 16-12 & -48+36 4-3 & -12+9 pmatrix = pmatrix 4 & -12 1 & -3 pmatrix = A B^2 = pmatrix -3 & 12 -1 & 4 pmatrix pmatrix -3 & 12 -1 & 4 pmatrix = pmatrix 9-12 & -36+48 3-4 & -12+16 pmatrix = pmatrix -3 & 12 -1 & 4 pmatrix = B Since A and B are idempotent matrices, A^k = A and B^k = B for all natural numbers k . The given