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JEE Main 2026 · 2 Apr · Shift 2, Mathematics question 22

Consider the matrices A=[2−24−2]A=\begin{bmatrix}2&-2\\4&-2\end{bmatrix} and B=[3913]B=\begin{bmatrix}3&9\\1&3\end{bmatrix}. If matrices P and Q are such that PA=BPA=B and AQ=BAQ=B, then the absolute value of the sum of the diagonal elements of 2(P+Q)2(P+Q) is ________.

Answer and worked solution

Official answer: 34

Matrices. P=BA−1P=BA^{-1} and Q=A−1BQ=A^{-1}B, and tr⁡(BA−1)=tr⁡(A−1B)\operatorname{tr}(BA^{-1})=\operatorname{tr}(A^{-1}B).

Compute. A−1=14[−22−42]A^{-1}=\frac14\begin{bmatrix}-2&2\\-4&2\end{bmatrix} and A−1B=14[−4−12−10−30]A^{-1}B=\frac14\begin{bmatrix}-4&-12\\-10&-30\end{bmatrix}, with trace −344-\frac{34}4.

Value. tr⁡2(P+Q)=4⋅(−344)=−34\operatorname{tr}2(P+Q)=4\cdot\left(-\frac{34}4\right)=-34; absolute value 34.

Answer: 34

Topics:Adjoint and inverseMatrix algebra
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