JEE Main 2026 — Apr 2, Shift 2 — Mathematics solution

JEE Main2026Apr 2, Shift 2Mathematics

Consider the matrices $A = [2 - 2 4 - 2]$ and $B = [3913]$ . If matrices $P$ and $Q$ are such that $P A = B$ and $A Q = B$ , then the absolute value of the sum of the diagonal elements of $2 (P + Q)$ is ______.

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Solution

$det A = 2 (- 2) - (- 2) (4) = - 4 + 8 = 4$ , so $A^{- 1} = \frac{1}{4} [- 22 - 42] = \frac{1}{2} [- 11 - 21]$ .

From $P A = B$ : $P = B A^{- 1} = \frac{1}{2} [3913] [- 11 - 21] = \frac{1}{2} [- 2112 - 74]$ .

From $A Q = B$ : $Q = A^{- 1} B = \frac{1}{2} [- 11 - 21] [3913] = \frac{1}{2} [- 2 - 6 - 5 - 15]$ .

Trace of $2 (P + Q) = 2 (tr P + tr Q) = 2 (\frac{- 21 + 4}{2} + \frac{- 2 - 15}{2}) = (- 17) + (- 17) = - 34$ .

Absolute value $= 34$ .

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