JEE Main 2025 Jan 29 Shift 2, Mathematics Q12: Algebra Of Matrices

JEE Main2025Jan 29, Shift 2Mathematics

Let $A = [a_{ij}]$ be a matrix of order $3 \times 3$ , 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

A
$280$
B
$168$
C
$210$
D
$224$

Solution

Write $a_{ij} = (2)^{i + j}$ entry by entry:

$A = 2224224424428$ .

The third row is $r_{3} = (4, 42, 8) = 4 (1, 2, 2)$ . The third row of $A^{2}$ equals $r_{3} A = 4 (1, 2, 2) A$ .

Compute $(1, 2, 2) A$ column-wise:

Column 1: $1 \cdot 2 + 2 \cdot 22 + 2 \cdot 4 = 2 + 4 + 8 = 14$ .

Column 2: $1 \cdot 22 + 2 \cdot 4 + 2 \cdot 42 = 22 + 42 + 82 = 142$ .

Column 3: $1 \cdot 4 + 2 \cdot 42 + 2 \cdot 8 = 4 + 8 + 16 = 28$ .

Third row of $A^{2} = 4 \cdot (14, 142, 28) = (56, 562, 112)$ .

Sum $= 56 + 112 + 562 = 168 + 562$ , so $α = 168$ , $β = 56$ .

$∴ α + β = 224$ .

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JEE Main 2025 Jan 29 Shift 2, Mathematics Q12: Algebra Of Matrices

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