<p>Let \( A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix} \) and \( B = A^{20} \). Then the sum of the elements of the first column of \( B \) is</p>
Step-by-Step Solution
Key Concept: Recognize that A is a lower triangular matrix with 1's on the diagonal. Computing A^n for lower triangular matrices follows a binomial pattern: the (i,j) entry of A^n equals C(n, i-j) when i ≥ j, and 0 otherwise.
<p><strong>Step 1:</strong> Recognize A is lower triangular with all diagonal entries = 1. For such matrices, A^n has a specific structure: the entry in position (i,j) where i ≥ j is C(n, i-j), and 0 otherwise.</p><p><strong>Step 2:</strong> For A^20, the first column contains entries at positions (1,1), (2,1), and (3,1):</p><p>• Position (1,1): C(20, 0) = 1</p><p>• Position (2,1): C(20, 1) = 20</p><p>• Position (3,1): C(20, 2) = (20×19)/2 = 190</p><p><strong>Step 3:</strong> Sum of first column = 1 + 20 + 190 = <strong>211</strong></p><p>∴ Answer: B</p>
Correct Answer: B