JEE Advanced 2025 · Paper 1, Mathematics question 4

Orthogonal integer matrices. Each row of $Q$ is an integer vector of length 1, so it is $\pm$ a standard basis vector. Hence $Q$ is a signed permutation matrix.

Commuting with $P$. $PQ=QP$ means $Q$ maps each eigenspace of $P$ to itself:

• the eigenspace for 2 is spanned by $e_1,e_2$;
• the eigenspace for 3 is spanned by $e_3$.

So $Q=\begin{pmatrix}A&0\\0&\pm1\end{pmatrix}$, where $A$ is a $2\times2$ signed permutation matrix. There are $2!\cdot2^2=8$ choices for $A$, giving
$$8\times2=16.$$