Let A = bmatrix 2 & -1 1 & 0 bmatrix and B = bmatrix -228 & 207 -206 & 184 bmatrix . If S = _ k=1 ²⁰ A^k and the system of equations (S + B) bmatrix x y bmatrix = bmatrix 0 0 bmatrix has a non-trivial solution, then which of the following relations holds true?

Step-by-step solution

Let A = I + N , where I = bmatrix 1 & 0 0 & 1 bmatrix and N = A - I = bmatrix 1 & -1 1 & -1 bmatrix . We can observe that N^2 = bmatrix 1 & -1 1 & -1 bmatrix bmatrix 1 & -1 1 & -1 bmatrix = bmatrix 0 & 0 0 & 0 bmatrix . Thus, A^k = (I + N)^k = I + kN . Now, we find the sum S = _ k=1 ²⁰ A^k = _ k=1 ²⁰ (I + kN) . S = 20I + N _ k=1 ²⁰ k = 20I + N ( 20 21 2 ) = 20I + 210N . Substituting I and N : S = bmatrix 20 & 0 0 & 20 bmatrix + bmatrix 210 & -210 210 & -210 bmatrix = bmatrix 230 & -210 210 & -190 bmatrix . We are g