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

Answer 1

Answer: (C)

$A=\begin{bmatrix}1&0&0\\ 1&1&0\\ 1&1&1\end{bmatrix}$
$A^{2}=\begin{bmatrix}1&0&0\\ 2&1&0\\ 3&2&1\end{bmatrix} ; A^{3}=\begin{bmatrix}1&0&0\\ 3&1&0\\ 6&3&1\end{bmatrix}$
$A^{4}=\begin{bmatrix}1&0&0\\ 4&1&0\\ 10&4&1\end{bmatrix}...... A^{20}=\begin{bmatrix}1&0&0\\ 20&1&0\\ 210&20&1\end{bmatrix}$
Sum of the elements of first column $= 231$