JEE Main 2026 Apr 2 Shift 2, Mathematics Q22: Algebra Of Matrices

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

From $PA=B$: $P=B{A}^{-1}=\frac{1}{2}\left[\begin{array}{c}39\\ 13\end{array}\right]\left[\begin{array}{c}-11\\ -21\end{array}\right]=\frac{1}{2}\left[\begin{array}{c}-2112\\ -74\end{array}\right]$.

From $AQ=B$: $Q=A^{-1}B=\frac{1}{2}\left[\begin{array}{c}-11\\ -21\end{array}\right]\left[\begin{array}{c}39\\ 13\end{array}\right]=\frac{1}{2}\left[\begin{array}{c}-2-6\\ -5-15\end{array}\right]$.

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

Absolute value $=34$.