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

JEE Advanced · 2016 · 22 May 2016 · Paper 2 · Matrices

Let

P = 1416014001

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

Answer: (B)

Write

P = A + I

where

A = 0416004000

. Compute

A^{2} = 0016000000

and

A^{3} = 0

. By the binomial theorem,

P^{50} = (I + A)^{50} = I + 50 A + (2 50) A^{2} = I + 50 A + 1225 A^{2}

. Since

Q = P^{50} - I

, we have

Q = 50 A + 1225 A^{2}

. Computing the entries:

q_{21} = 50 \cdot 4 = 200

q_{31} = 50 \cdot 16 + 1225 \cdot 16 = 16 (50 + 1225) = 16 \cdot 1275 = 20400

q_{32} = 50 \cdot 4 = 200

. Therefore

\frac{q _{31} + q _{32}}{q _{21}} = \frac{20400 + 200}{200} = \frac{20600}{200} = 103

JEE Advanced · 2016 · 22 May 2016 · Paper 2 · Single correct option