JEE Advanced 2025 Paper 1 Q04 Mathematics Matrices & Determinants Matrix Operations Hard

JEE Advanced 2025 Paper 1 · Q04 · Matrix Operations

Consider the matrix $P = \text{diag}(2, 2, 3)$ (i.e., $3\times 3$ with diagonal entries $2, 2, 3$ and zeros elsewhere). Find the number of $3\times 3$ invertible matrices $Q$ with integer entries such that $Q^{-1} = Q^T$ and $PQ = QP$.

  1. A. 32
  2. B. 8
  3. C. 16
  4. D. 24
Reveal answer + step-by-step solution

Correct answer:C

Solution

$Q^{-1} = Q^T$ means $Q$ is orthogonal: rows are orthonormal. With integer entries, each row must have a single non-zero entry equal to $\pm 1$ (any other integer choice would have norm $> 1$, contradicting unit length). So $Q$ is a signed permutation matrix. $PQ = QP$: writing $P = \text{diag}(2,2,3)$, the $(i,j)$ entry of $PQ$ is $p_i Q_{ij}$ and of $QP$ is $p_j Q_{ij}$. So $Q_{ij} \ne 0$ only when $p_i = p_j$, i.e., $(i,j)$ lies in the $(1,2)$-block or at $(3,3)$. Hence $Q$ is block-diagonal with a $2\times 2$ signed permutation in the top-left and a $\pm 1$ at $(3,3)$. Count: $2\times 2$ signed permutations $= 2! \cdot 2^2 = 8$. Times $2$ for the $(3,3)$ sign $= 16$. Answer: $16$.

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