Matrices & Determinants · JEE Advanced previous year questions

2025 2 questions

1. Q42025 · 18 May 2025 · Paper 1

   Consider the matrix $$P=\left(\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right).$$ Let the transpose of a matrix $X$ be denoted by $X^T$. Then the number of $3\times 3$ invertible matrices $Q$ with integer entries, such that $$Q^{-1}=Q^T\text{and}PQ=QP,$$ is

   1. A.32
   2. B.8
   3. C.16
   4. D.24
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   Answer: C

2. Q52025 · 18 May 2025 · Paper 2· One or more correct

   Let $I=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ and $P=\left(\begin{array}{cc}2& 0\\ 0& 3\end{array}\right)$. Let $Q=\left(\begin{array}{cc}x& y\\ z& 4\end{array}\right)$ for some non-zero real numbers $x$, $y$, and $z$, for which there is a $2\times 2$ matrix $R$ with all entries being non-zero real numbers, such that $QR=RP$. Then which of the following statements is (are) TRUE?

   1. A.The determinant of $Q-2I$ is zero
   2. B.The determinant of $Q-6I$ is 12
   3. C.The determinant of $Q-3I$ is 15
   4. D.$yz=2$
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   Answer: A, B