Consider the matrix P = ( 2 0 0 0 2 0 0 0 3 ) . 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 [2025]

Question

Consider the matrix $P = (\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}) .$

Let the transpose of a matrix $X$ be denoted by $X^{T}$ . Then the number of $3 \times 3$ invertible matrices $Q$ , with integer entries, such that $Q^{- 1} = Q^{T}$ and $P Q = Q P,$ is [2025]

Smart Achievers · Accepted Answer

(3)

$Q^{- 1} = Q^{T} \Rightarrow Q Q^{T} = I$ $(given)$

Hence, $Q$ is an orthogonal matrix.

Let $Q = [\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}]$

$P Q = Q P$

$\Rightarrow [\begin{matrix} 2 a_{1} & 2 b_{1} & 2 c_{1} \\ 2 a_{2} & 2 b_{2} & 2 c_{2} \\ 3 a_{3} & 3 b_{3} & 3 c_{3} \end{matrix}] = [\begin{matrix} 2 a_{1} & 2 b_{1} & 3 c_{1} \\ 2 a_{2} & 2 b_{2} & 3 c_{2} \\ 2 a_{3} & 2 b_{3} & 3 c_{3} \end{matrix}]$

$\Rightarrow c_{1} = 0,$ $c_{2} = 0,$ $a_{3} = 0,$ $b_{3} = 0$

So, $Q = [\begin{matrix} a_{1} & b_{1} & 0 \\ a_{2} & b_{2} & 0 \\ 0 & 0 & c_{3} \end{matrix}]$

Since $Q$ is orthogonal,

$a_{1} a_{2} + b_{1} b_{2} = 0,$ $a_{1}^{2} + b_{1}^{2} = 1,$ $a_{2}^{2} + b_{2}^{2} = 1,$ $c_{3}^{2} = 1$

$\begin{matrix} a_{1} & b_{1} & a_{2} & b_{2} & c_{3} \\ 1 & 0 & 0 & 1, - 1 & 1, - 1 \\ - 1 & 0 & 0 & 1, - 1 & 1, - 1 \\ 0 & 1 & 1, - 1 & 0 & 1, - 1 \\ 0 & - 1 & 1, - 1 & 0 & 1, - 1 \end{matrix}$

$∴$ $Total 16 matrices$

Solution

Q.
Consider the matrix $P = (\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}) .$

Let the transpose of a matrix $X$ be denoted by $X^{T}$ . Then the number of $3 \times 3$ invertible matrices $Q$ , with integer entries, such that $Q^{- 1} = Q^{T}$ and $P Q = Q P,$ is [2025]

1 32

2 8

3 16

4 24

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