Let A = [ a ij ] be a matrix of order 3 × 3 , with a ij =(√(2)) i + j . If the sum of all the elements in the third row of A 2 is α+β √(2), α, β in Z , then α+β is equal to

Mathematics · Matrices

JEE Main 2025 — 29 January, Evening Shift — Question 53

Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]$ be a matrix of order $3 \times 3$ , with $\mathrm{a}_{\mathrm{ij}}=(\sqrt{2})^{\mathrm{i}+\mathrm{j}}$ .

If the sum of all the elements in the third row of $\mathrm{A}^{2}$ is $\alpha+\beta \sqrt{2}, \alpha, \beta \in \mathbf{Z}$ , then $\alpha+\beta$ is equal to

Option A:
280
Option B:
168
Option C:
210
Option D:
224
Correct

Answer: D

Step-by-step solution

$\quad A=\left[\begin{array}{lll}(\sqrt{2})^{2} & (\sqrt{2})^{3} & (\sqrt{2})^{4} \\ (\sqrt{2})^{3} & (\sqrt{2})^{4} & (\sqrt{2})^{5} \\ (\sqrt{2})^{4} & (\sqrt{2})^{5} & (\sqrt{2})^{6}\end{array}\right]$

$A=\left[\begin{array}{ccc}2 & 2 \sqrt{2} & 4 \\ 2 \sqrt{2} & 4 & 4 \sqrt{2} \\ 4 & 4 \sqrt{2} & 8\end{array}\right]$

$A^{2}=2^{2}\left[\begin{array}{ccc}1 & \sqrt{2} & 2 \\ \sqrt{2} & 2 & 2 \sqrt{2} \\ 2 & 2 \sqrt{2} & 4\end{array}\right]\left[\begin{array}{ccc}1 & \sqrt{2} & 2 \\ \sqrt{2} & 2 & 2 \sqrt{2} \\ 2 & 2 \sqrt{2} & 4\end{array}\right]$

$=4\left[\begin{array}{ccc}- & - & - \\ - & - & - \\ (2+4+8) & (2 \sqrt{2}+4 \sqrt{2}+8 \sqrt{2}) & (4+8+16)\end{array}\right]$

Sum of elements of $3^{\text {rd }}$ row $=4(14+14 \sqrt{2}+28)$

$\begin{aligned} = & 4(42+14 \sqrt{2}) \\ = & 168+56 \sqrt{2} \\ & \alpha+\beta \sqrt{2} \\ \therefore \alpha & \alpha \beta=168+56=224 \end{aligned}$

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 29 January, Evening Shift
Subject: Mathematics
Chapter: Matrices
Topic: Types of matrices & its properties