1
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 XT. Then the number of 3 3 invertible matrices Q with integer entries, such that Q-1 = QT and PQ = QP, is
- 32
- 8
- 16
- 24
Answer
(3) 16
Solution
Q-1 = QT means Q is orthogonal: each row is a unit vector. With integer entries, each row has exactly one entry 1, the rest 0 (signed permutation matrix).
PQ = QP: comparing entries, pi qij = qij pj, so qij = 0 whenever pi pj, i.e. q13 = q23 = q31 = q32 = 0.
So Q = bmatrix A & 0 \ 0 & 1 bmatrix with A a 2 2 signed permutation matrix.
Number of A: 2 permutations 22 sign choices = 8; last entry: 2 choices.
Total = 8 2 = 16.
Answer: (C)
JEE Advanced 2025 Paper 1