Question

Let $P =\begin{bmatrix}1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1\end{bmatrix}$ and $I$ be the identity matrix of order 3 . If $Q =\left[ q _{ij}\right]$ is a matrix such that $P ^{50}- Q = I$, then $\frac{ q _{31}+ q _{32}}{ q _{21}}$ equals

• A. 52
• B. 103
• C. 201
• D. 205

Answer 1

Answer: (B)

$P ^{2}=\begin{bmatrix}1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1\end{bmatrix}$
$P ^{2}=\begin{bmatrix}1 & 0 & 0 \\ 4+4 & 1 & 0 \\ 4^{2}+4^{2}+16 & 0+4+4 & 1\end{bmatrix}$
$ P ^{3}=\begin{bmatrix}1 & 0 & 0 \\8 & 1 & 0 \\48 & 8 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 \\4 & 1 & 0 \\16 & 4 & 1\end{bmatrix}$
$=\begin{bmatrix}1 & 0 & 0 \\8+4 & 1 & 0 \\48+32+16 & 8+4 & 1 \end{bmatrix}$
$\therefore P ^{ n }=\begin{bmatrix}1 & 0 & 0 \\4 n & 1 & 0 \\\frac{ n ( n +1)}{2} 16 & 4 n & 1\end{bmatrix}$
Now, $Q=P^{50}-I$
$Q=\begin{bmatrix}1 & 0 & 0 \\200 & 1 & 0 \\20400 & 200 & 1\end{bmatrix}-\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$
$=\begin{bmatrix}0 & 0 & 0 \\200 & 0 & 0 \\20400 & 200 & 0\end{bmatrix}$
$\therefore \frac{q_{31}+q_{32}}{q_{21}}=\frac{20400+200}{200}=103$