JEE Advanced · Mathematics · 18. Matrices

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

A $52$
B $103$
C $201$
D $205$

Verified Solution

Answer & Solution

Correct Answer

(B) $103$

Step-by-step Solution

Detailed explanation

P = [\begin{matrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{matrix}] \Rightarrow P^{2} = [\begin{matrix} 1 & 0 & 0 \\ 8 & 1 & 0 \\ 16 + 32 & 8 & 1 \end{matrix}]

So,

P^{3} = [\begin{matrix} 1 & 0 & 0 \\ 12 & 1 & 0 \\ 16 + 32 + 48 & 12 & 1 \end{matrix}]

Using symmetry, we can write

P^{50} = [\begin{matrix} 1 & 0 & 0 \\ 200 & 1 & 0 \\ \frac{16.50.51}{2} & 200 & 1 \end{matrix}]

As,

P^{50} - Q = I \Rightarrow Q = P^{50} - I = [\begin{matrix} 0 & 0 & 0 \\ 200 & 0 & 0 \\ \frac{16.50.51}{2} & 200 & 0 \end{matrix}],

q_{31} = \frac{16.50.51}{2},

q_{32} = 200

and

q_{21} = 200

∴

\frac{q_{31} + q_{32}}{q_{21}} = \frac{16.50.51}{2.200} + 1 = 102 + 1 = 103

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

	List- I		List- II
$(I)$	$2 h + k$	$(P)$	$6$
$(I I)$	$\frac{L e n g t h o f Z W}{L e n g t h o f X Y}$	$(Q)$	$\sqrt{6}$
$(I I I)$	$\frac{A r e a o f t r i a n g l e M Z N}{A r e a o f t r i a n g l e Z M W}$	$(R)$	$\frac{5}{4}$
$(I V)$	$α$	$(S)$	$\frac{21}{5}$
		$(T)$	$2 \sqrt{6}$
		$(U)$	$\frac{10}{3}$

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Let P=1004101641 and I be the identity matrix of order 3. If Q=qij is a matrix such that P50-Q=I, then the value of q31+q32q21 is equal to

Step-by-step Solution

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