If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + (A^T)^{20} is equal to
Step-by-Step Solution
Key Concept: The matrix A is a lower triangular matrix. Calculate powers of A and A^T to find the sum.
Given A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}. This is a lower triangular matrix. Calculating powers of A reveals a pattern. A^2 = \begin{pmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 10 & 4 & 1 \end{pmatrix}. Continuing this, A^n can be determined. The sum A^{20} + (A^T)^{20} simplifies based on the properties of these specific matrices.
Correct Answer: B