Consider the matrix, P= ( array lll 2 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 3 array ) 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 P Q=Q P is

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Quantrex Academy · Accepted Answer

Correct answer: C. 16

$ aligned & PQ = QP [ array ccc 2 a _1 & 2 ~b _1 & 2 c _1 \ 2 a _2 & 2 ~b _2 & 2 c _2 \ 3 a _3 & 3 ~b _3 & 3 c _3 array ]= [ array ccc 2 a _1 & 2 ~b _1 & 3 c _1 \ 2 a _2 & 2 ~b _2 & 3 c _2 \ 2 a _3 & 2 ~b _3 & 3 c _3 array ] \ & c _1=0, c _2=0, a _3=0, ~b _3=0 \ & Q = [ array ccc a _1 & ~b _1 & 0 \ a _2 & ~b _2 & 0 \ 0 & 0 & c _3 array ] \ & a _1 a _2+ b _1 ~b _2=0 \ & a _1^2+ b _1^2=1, a _2^2+ b _2^2=1, c _3^2=1 aligned $ $ array lllll a_1 & b_1 & a_2 & b_2 & c_3 \ 1 & 0 & 0 & 1,-1 & +1,-1 \ -1 & 0 & 0 & 1,-1 & 1,-1 \ 0 & 1 & 1,-1 & 0 & 1,-1 \ 0 & -1 & 1,-1 & 0 & 1,-1 array $ Total 16 matrices

JEE Advanced Mathematics Question (2025) — Solution

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