If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + A^{19} + ... + A + I = ?
Step-by-Step Solution
Key Concept: The matrix A is a lower triangular matrix. By calculating powers of A, we can identify a pattern and use the formula for the sum of a geometric series of matrices.
Given A = I + N, where N = [0 0 0; 2 0 0; 3 2 0]. N^2 = [0 0 0; 0 0 0; 4 0 0], N^3 = 0. Using binomial expansion for A^k = (I+N)^k = I + kN + k(k-1)/2 * N^2. Summing from k=0 to 20 gives the result.
Correct Answer: 1