<p>Given <br/> \(A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}\), \(B = A^{20}\)<br/> Find the sum of elements of the first column of \(B\).</p>
Step-by-Step Solution
Key Concept: A is a lower triangular matrix with 1s on the diagonal. For such matrices, A^n follows a binomial expansion pattern where the (i,j) entry equals C(n, i-j) for i ≥ j. Use this pattern to find the first column of A^20 directly.
<p><strong>Step 1:</strong> Recognize that A is a lower triangular matrix with all diagonal entries equal to 1. For such matrices, the (i,j) entry of A^n is given by:</p><p>(A^n)_{i,j} = C(n, i-j) for i ≥ j, and 0 for i < j</p><p><strong>Step 2:</strong> Apply this formula to find the first column (j=1) of A^20:</p><p>• (A^20)_{1,1} = C(20, 1-1) = C(20, 0) = 1</p><p>• (A^20)_{2,1} = C(20, 2-1) = C(20, 1) = 20</p><p>• (A^20)_{3,1} = C(20, 3-1) = C(20, 2) = 190</p><p><strong>Step 3:</strong> Sum the elements of the first column:</p><p>Sum = 1 + 20 + 190 = 211</p><p>∴ Answer: C</p>
Correct Answer: C