Consider the matrices A = [ begin array cc 2 & -2\\ 4 & -2 end array ] and B = [ begin array cc 3 & 9\\ 1 & 3 end array ] . If matrices P and Q are such that PA = B and AQ = B , then the absolute value of the sum of the diagonal elements of 2( P + Q ) is

Mathematics · Matrices

JEE Main 2026 — 2 April, Evening Shift — Question 44

Consider the matrices $\mathrm{A} = \left[ \begin{array}{cc}2 & -2\\ 4 & -2 \end{array} \right]$ and $\mathrm{B} = \left[ \begin{array}{cc}3 & 9\\ 1 & 3 \end{array} \right]$ . If matrices $\mathbf{P}$ and $\mathbf{Q}$ are such that $\mathbf{PA} = \mathbf{B}$ and $\mathbf{AQ} = \mathbf{B}$ , then the absolute value of the sum of the diagonal elements of $2(\mathbf{P} + \mathbf{Q})$ is

Answer: 34

Numerical answer — enter this value.

Step-by-step solution

$A^{-1}=\frac{1}{4}\left[\begin{array}{ll}-2 & 2\\ -4 & 2\end{array}\right]=\frac{1}{2}\left[\begin{array}{ll}-1 & 1\\ -2 & 1\end{array}\right]$

\begin{aligned} & \because \mathrm{PA}=\mathrm{B} \\& \Rightarrow \mathrm{P}=\mathrm{BA}^{-1} \\& =\frac{1}{2}\left[\begin{array}{ll} 3 & 9 \\ 1 & 3 \end{array}\right]\left[\begin{array}{ll} -1 & 1 \\ -2 & 1 \end{array}\right]=\frac{1}{2}\left[\begin{array}{cc} -21 & 12 \\ -7 & 4 \end{array}\right] \\& \because \mathrm{AQ}=\mathrm{B} \\& \Rightarrow \mathrm{Q}=\mathrm{A}^{-1} \mathrm{~B} \\& =\frac{1}{2}\left[\begin{array}{ll} -1 & 1 \\ -2 & 1 \end{array}\right]\left[\begin{array}{ll} 3 & 9 \\ 1 & 3 \end{array}\right]=\frac{1}{2}\left[\begin{array}{cc} -2 & -6 \\ -5 & -15 \end{array}\right] \\& \therefore|\operatorname{tr}(2(\mathrm{P}+\mathrm{Q}))|=|(-21)+(4)+(-2)+(-15)| \\& =|-34|=34 \end{aligned}

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 2 April, Evening Shift
Subject: Mathematics
Chapter: Matrices
Topic: solving System of Linear Equations using Matrices