Question

Let $P=\left[\begin{array}{ccc}1& 0& 0\\ 4& 1& 0\\ 16& 4& 1\end{array}\right]$ 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)

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