Let A = bmatrix 1 & 2 & 3 0 & 1 & 2 0 & 0 & 1 bmatrix and A^k = [a_ ij ] for some positive integer k . If the element a₁₃ = 300 , then the value of a₁₂ + a₂₃ is :

Step-by-step solution

Let A = I + N , where I is the identity matrix and N = bmatrix 0 & 2 & 3 0 & 0 & 2 0 & 0 & 0 bmatrix . Calculating the powers of N : N^2 = bmatrix 0 & 2 & 3 0 & 0 & 2 0 & 0 & 0 bmatrix bmatrix 0 & 2 & 3 0 & 0 & 2 0 & 0 & 0 bmatrix = bmatrix 0 & 0 & 4 0 & 0 & 0 0 & 0 & 0 bmatrix N^3 = bmatrix 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 bmatrix Using the binomial expansion for A^k since I and N commute: A^k = (I + N)^k = I + kN + k(k-1) 2 N^2 Substituting the matrices N and N^2 : A^k = bmatrix 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 bmatrix