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

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 2025Paper 1

Exam: JEE Advanced
Year: 2025
Paper: Paper 1
Subject: Mathematics
Chapter: Matrices

JEE Advanced 2025 · Paper 1 · Mathematics · Matrices

Consider the matrix, P= ( array lll 2 & 0 & 0 0 & 2 & 0 0 & 0 & 3...

Consider the matrix,

P=

\left(

\begin{array}{lll}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{array}

\right)

Let the transpose of a matrix

be denoted by

\mathrm{X}^T

. Then the number of

3 \times 3

invertible matrices Q with integer entries, such that

Q^{-1}=Q^T and

P Q=Q P

Options

Correct answer

C. 16

Step-by-step solution

$\begin{aligned} \mathrm{PQ}=\mathrm{QP} \Rightarrow\left[\begin{array}{ccc}2 \mathrm{a}_{1} & 2 \mathrm{ b}}_{1} & 2 \mathrm{c}_{1} \\ 2 \mathrm{a}}_{2} & 2 \mathrm{ b}_{2} & 2 \mathrm{c}}_{2} \\ 3 \mathrm{a}_{3} & 3 \mathrm{ b}}_{3} & 3 \mathrm{c}_{3} \end{array} \right]=\left[ \begin{array}{ccc}2 \mathrm{a}}_{1} & 2 \mathrm{ b}_{1} & 3 \mathrm{c}}_{1} \\ 2 \mathrm{a}_{2} & 2 \mathrm{ b}}_{2} & 3 \mathrm{c}_{2} \\ 2 \mathrm{a}}_{3} & 2 \mathrm{ b}_{3} & 3 \mathrm{c}}_{3} \end{array} \right] \\ \mathrm{c}_{1}=0, \mathrm{c}}_{2}=0, \mathrm{a}_{3}=0, \mathrm{ b}}_{3}=0 \\ \mathrm{Q}=\left[\begin{array}{ccc} \mathrm{a}_{1} & \mathrm{ b}}_{1} & 0 \\ \mathrm{a}_{2} & \mathrm{ b}}_{2} & 0 \\ 0 & 0 \mathrm{c}_{3} \end{array}\right] \\ & \mathrm{a}}_{1} \mathrm{a}_{2}+ \mathrm{b}}_{1} \mathrm{ b}_{2}=0 \\ & \mathrm{a}}_{1}^2+\mathrm{b}_{1}^2=1, \mathrm{a}}_{2}^2+\mathrm{b}_{2}^2=1, \mathrm{c}}_{3}^2=1 \end{aligned}

)

\begin{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 \end{array}

$$) Total 16 matrices

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