Let A = pmatrix 0 & 1 & 0 0 & 0 & 1 1 & 0 & 0 pmatrix and B = pmatrix 0 & 0 & 1 1 & 0 & 0 0 & 1 & 0 pmatrix . If M_n = A^n + B^n for n N , and S = _ n=1 ²⁰ M_n , then the value of (S) is

Step-by-step solution

We have A = pmatrix 0 & 1 & 0 0 & 0 & 1 1 & 0 & 0 pmatrix . By matrix multiplication, A^2 = pmatrix 0 & 0 & 1 1 & 0 & 0 0 & 1 & 0 pmatrix = B , and A^3 = I . Since B = A^2 , we have B^2 = A^4 = A and B^3 = A^6 = I . Let us evaluate the first few terms of the sequence M_n = A^n + B^n : M₁ = A + B M₂ = A^2 + B^2 = B + A = A + B M₃ = A^3 + B^3 = I + I = 2I The sequence M_n is periodic with period 3. The sum of one period is: M₁ + M₂ + M₃ = 2(A + B + I) Let J = A + B + I = pmatrix 1 & 1 & 1 1 & 1 & 1 1 & 1 & 1 pmatrix