Consider the matrix P = pmatrix 2 & 0 & 0 0 & 2 & 0 0 & 0 & 3 pmatrix . Let the transpose of a matrix X be denoted by X^T. Then the number of 3 × 3 invertible matrices Q with integer entries, such that Q^ -1 = Q^T and PQ = QP, is

JEE Advanced 2025 Mathematics question

Consider the matrix

P = .

Let the transpose of a matrix X be denoted by . Then the number of 3 × 3 invertible matrices Q with integer entries, such that

= and PQ = QP,

Answer: (C)

Comparing entries of PQ and QP: , so whenever . Thus Q is block diagonal: a 2 × 2 block (rows/columns 1, 2) and a 1 × 1 block. Q is orthogonal with

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Question ID VK-037791