JEE Main 2025 Jan 22 Shift 2, Mathematics Q4: Algebra Of Matrices

JEE Main2025Jan 22, Shift 2Mathematics

For a $3 \times 3$ matrix $M$ , let trace $(M)$ denote the sum of all the diagonal elements of $M$ . Let $A$ be a $3 \times 3$ matrix such that $∣ A ∣ = \frac{1}{2}$ and trace $(A) = 3$ . If $B = adj (adj (2 A))$ , then the value of $∣ B ∣ + trace (B)$ equals :

A
56
B
132
C
174
D
280

Solution

For an $n \times n$ matrix $M$ , $adj (adj (M)) = ∣ M ∣^{n - 2} M$ . With $n = 3$ , this gives $adj (adj (M)) = ∣ M ∣ M$ .

Apply to $M = 2 A$ :

$B = adj (adj (2 A)) = ∣2 A ∣ \cdot (2 A) = 2^{3} ∣ A ∣ \cdot 2 A = 8 \cdot \frac{1}{2} \cdot 2 A = 8 A$ .

Now compute the required quantities:

$∣ B ∣ = ∣8 A ∣ = 8^{3} ∣ A ∣ = 512 \cdot \frac{1}{2} = 256$ .

$trace (B) = 8 \cdot trace (A) = 8 \cdot 3 = 24$ .

$∴ ∣ B ∣ + trace (B) = 256 + 24 = 280$ .

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JEE Main 2025 Jan 22 Shift 2, Mathematics Q4: Algebra Of Matrices

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