Let A = bmatrix 4 & -9 1 & -2 bmatrix and B = bmatrix -30 & 91 -10 & 30 bmatrix . If the system of linear equations (A^k + B) bmatrix x y bmatrix = bmatrix 0 0 bmatrix has a non-trivial solution for some positive integer k , then the value of k is

Step-by-step solution

Let A = I + N , where I = bmatrix 1 & 0 0 & 1 bmatrix and N = A - I = bmatrix 3 & -9 1 & -3 bmatrix . Now, N^2 = bmatrix 3 & -9 1 & -3 bmatrix bmatrix 3 & -9 1 & -3 bmatrix = bmatrix 0 & 0 0 & 0 bmatrix . Since N^2 = 0 , we can use the binomial expansion to find A^k : A^k = (I + N)^k = I + kN = bmatrix 1 & 0 0 & 1 bmatrix + bmatrix 3k & -9k k & -3k bmatrix = bmatrix 1+3k & -9k k & 1-3k bmatrix . We are given B = bmatrix -30 & 91 -10 & 30 bmatrix . Then, A^k + B = bmatrix 3k-29 & 91-9k k-10 & 31-3k bmatrix . For the