Question

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

103

$P=\left[\begin{array}{ccc}1& 0& 0\\ 4& 1& 0\\ 16& 4& 1\end{array}\right]\Rightarrow P^2=\left[\begin{array}{ccc}1& 0& 0\\ 8& 1& 0\\ 16(1+2)& 8& 1\end{array}\right],P^3=\left[\begin{array}{ccc}1& 0& 0\\ 12& 1& 0\\ 16(1+2+3)& 12& 1\end{array}\right]$

$P^{50}=\left[\begin{array}{ccc}1& 0& 0\\ 4\times 50& 1& 0\\ 16(1+2+..50)& 4\times 50& 1\end{array}\right],P^{50}=\left[\begin{array}{ccc}1& 0& 0\\ 200& 1& 0\\ 20400& 200& 1\end{array}\right]\because P^{50}-Q=I$

$\Rightarrow \left[\begin{array}{ccc}1& 0& 0\\ 200& 1& 0\\ 20400& 200& 1\end{array}\right]-\left[\begin{array}{ccc}{q}_{11}& q_{12}& q_{13}\\ q_{21}& q_{22}& q_{23}\\ q_{31}& q_{32}& q_{33}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 0& 1\end{array}\right]$

$\Rightarrow 200-q_{21}=0⟹q_{21}=200$

$\Rightarrow 20400-q_{31}=0$

$\Rightarrow q_{31}=20400$ and $200-q_{32}=0$

$\Rightarrow q_{32}=200$

$\therefore \frac{q_{31}+q_{32}}{q_{21}}=\frac{20400+200}{200}=103$