Question

Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 1& 1& 1\end{array}\right]$ 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)

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