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 $Q = [q_{ij}]$ is a matrix such that $P^{50} - Q = I$, then $\frac{q_{31} + q_{32}}{q_{21}}$ equals

P = 1 0 0 4 1 0 16 4 1 , I = I₃

$P^{50} - Q = I \Rightarrow Q = P^{50} - I$

$\text{Find } \dfrac{q_{31}+q_{32}}{q_{21}}$

$\text{so we need } P^{50}$

$P = I + N,\qquad N = P - I$

$\text{If } N^{k}=0 \text{ for some } k, \text{ expand } (I+N)^{50}$

N = P - I = 0 0 0 4 0 0 16 4 0 N² = 0 0 0 0 0 0 16 0 0

$N^{3} = N^{2}\cdot N = 0$

$P^{50} = (I+N)^{50}$

$= I + 50N + \binom{50}{2}N^{2}$

50N = 0 0 0 200 0 0 800 200 0 , binom502=1225 1225 N² = 0 0 0 0 0 0 19600 0 0 P⁵⁰ = 1 0 0 200 1 0 20400 200 1 Q = P⁵⁰-I = 0 0 0 200 0 0 20400 200 0

$\dfrac{q_{31}+q_{32}}{q_{21}} = \dfrac{20400+200}{200} = \dfrac{20600}{200} = 103$

$\boxed{\text{Answer: } 103 \text{ — option (2)}}$