SATHEE: Matrices and Determinants 1 Question 6

Matrices and Determinants 1 Question 6

«««< HEAD

6. Let $P = \begin{array}{rrr} 1 & 0 & 0 4 & 1 & 0 \end{array}$ and $I$ be the identity matrix of order 3 . $\begin{array}{lll} 16 & 4 & 1 \end{array}$ If $Q = [q_{i j}]$ is a matrix, such that $P^{50} - Q = I$ , then $\frac{q_{31} + q_{32}}{q_{21}}$ equals

======= ####6. Let $P = [\begin{array}{ccc} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{array}]$ and $I$ be the identity matrix of order 3 . If $Q = [q_{i j}]$ is a matrix, such that $P^{50} - Q = I$ , then $\frac{q_{31} + q_{32}}{q_{21}}$ equals

3e0f7ab6f6a50373c3f2dbda6ca2533482a77bed

(2016 Adv.)

(a) 52

(b) 103

(d) 205

Show Answer

Answer:

Correct Answer: 6. (b)

Solution:

Here,

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

$\begin{aligned} ∴ P^{2} & = [\begin{array}{ccc} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{array}] [\begin{array}{ccc} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{array}] = [\begin{array}{ccc} 1 & 0 & 0 \\ 4 + 4 & 1 & 0 \\ 16 + 32 & 4 + 4 & 1 \end{array}] \\ = [\begin{array}{ccc} 1 & 0 & 0 \\ 4 \times 2 & 1 & 0 \\ 16 (1 + 2) & 4 \times 2 & 1 \end{array}] \\ and P^{3} & = [\begin{array}{ccc} 1 & 0 & 0 \\ 4 \times 2 & 1 & 0 \\ 16 (1 + 2) & 4 \times 2 & 1 \end{array}] [\begin{array}{ccc} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{array}] \\ = [\begin{array}{ccc} 1 & 0 & 0 \\ 4 \times 3 & 1 & 0 \\ 16 (1 + 2 + 3) & 4 \times 3 & 1 \end{array}] \end{aligned}$

From symmetry,

$\begin{aligned} P^{50} = [\begin{array}{ccc} 1 & 0 & 0 \\ 4 \times 50 & 1 & 0 \\ 16 (1 + 2 + 3 + \dots + 50) & 4 \times 50 & 1 \end{array}] \\ ∵ P^{50} - Q = I \\ [given] \\ ∴ [\begin{array}{ccc} 1 - q_{11} & - q_{12} & - q_{13} \\ 200 - q_{21} & 1 - q_{22} & - q_{23} \\ 16 \times \frac{50}{2} (51) - q_{31} & 200 - q_{32} & 1 - q_{33} \end{array}] = [\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] \\ \Rightarrow 200 - q_{21} = 0, \frac{16 \times 50 \times 51}{2} - q_{31} = 0, \\ 200 - q_{32} = 0 \\ ∴ q_{21} = 200, q_{32} = 200, q_{31} = 20400 \end{aligned}$