If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + (adj A)^{20} is equal to
Step-by-Step Solution
Key Concept: Calculate A^2, A^3 to find a pattern for A^n, then find adj A and its powers.
Given A = [[1,0,0],[2,1,0],[3,2,1]]. A is a lower triangular matrix. A^2 = [[1,0,0],[4,1,0],[12,4,1]]. A^n = [[1,0,0],[2n,1,0],[n(2n+1),2n,1]]. adj A = [[1,0,0],[-2,1,0],[1,-2,1]]. (adj A)^n = [[1,0,0],[-2n,1,0],[n(2n-1),-2n,1]]. Summing these gives the result.
Correct Answer: 4