Let A = bmatrix 0 & 1 & 0 0 & 0 & 1 1 & 0 & 0 bmatrix and B₀ = A²⁰ + 2A²¹ + 3A²² . If B_n = adj (B_ n-1 ) for all n 1 , then (B₂) is equal to

Step-by-step solution

We are given A = bmatrix 0 & 1 & 0 0 & 0 & 1 1 & 0 & 0 bmatrix . Let us compute the powers of A : A^2 = bmatrix 0 & 1 & 0 0 & 0 & 1 1 & 0 & 0 bmatrix bmatrix 0 & 1 & 0 0 & 0 & 1 1 & 0 & 0 bmatrix = bmatrix 0 & 0 & 1 1 & 0 & 0 0 & 1 & 0 bmatrix A^3 = A^2 A = bmatrix 0 & 0 & 1 1 & 0 & 0 0 & 1 & 0 bmatrix bmatrix 0 & 1 & 0 0 & 0 & 1 1 & 0 & 0 bmatrix = bmatrix 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 bmatrix = I Since A^3 = I , we can reduce the exponents in B₀ modulo 3: A²⁰ = (A^3)^6 A^2 = A^2 A²¹ = (A^3)^7 = I A²² = (A^3)^7 A