Question

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 :

• A. 210
• B. 211
• C. 231
• D. 251

Hint:

Compute \(A^2\) and \(A^3\) to see the pattern: the first column of \(A^n\) is \((1, n, \frac{n(n+1)}{2})^T\).

Answer 1

The Correct Option is C

\(A^2 = \begin{bmatrix}1&0&0\\2&1&0\\3&2&1\end{bmatrix}\), \(A^3 = \begin{bmatrix}1&0&0\\3&1&0\\6&3&1\end{bmatrix}\).
The first column of \(A^n\) is \(\left(1,\ n,\ \frac{n(n+1)}{2}\right)\) (valid for n = 1, 2, 3 and follows by induction since multiplying by A adds the previous entries).
For n = 20: entries \(1,\ 20,\ \frac{20\cdot21}{2} = 210\).
Sum \(= 1 + 20 + 210 = 231\).
Answer: (3) 231