Math JEE Adv. 2025 paper 1

 

To solve the problem, we need to find the number of $3\times 3$ invertible integer matrices $Q$ that satisfy the following conditions:

1. Orthogonality: $Q^{-1}=Q^T$, which implies that $Q$ is an orthogonal matrix.

2. Commutation with $P$: $PQ=QP$, where $P$ is the given diagonal matrix.

Step 1: Analyze the Matrix $P$

The matrix $P$ is given by:

$$P=\left(\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right)\mathrm{.}$$

This is a diagonal matrix with distinct eigenvalues $2$ (with multiplicity 2) and $3$.

Step 2: Commutation Condition $PQ=QP$

For a matrix $Q$ to commute with $P$, it must preserve the eigenspaces of $P$. Since $P$ has two distinct eigenvalues (2 and 3), $Q$ must be block-diagonal with respect to these eigenspaces. Specifically:

• The first two rows and columns of $Q$ correspond to the eigenvalue 2.

• The third row and column correspond to the eigenvalue 3.

Thus, $Q$ must have the form:

$$Q=\left(\begin{array}{cc}A& 0\\ 0& b\end{array}\right),$$

where:

• $A$ is a $2\times 2$ orthogonal matrix (since $Q$ is orthogonal).

• $b$ is a scalar such that $b^2=1$ (to preserve orthogonality), implying $b=\pm 1$.

Step 3: Orthogonal $2\times 2$ Matrices with Integer Entries

The $2\times 2$ orthogonal matrices with integer entries are precisely the signed permutation matrices (i.e., matrices with exactly one non-zero entry of $\pm 1$ in each row and column). There are 8 such matrices:

1. The identity matrix:

   $$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\mathrm{.}$$

2. The negation of the identity matrix:

   $$\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)\mathrm{.}$$

3. The two permutation matrices:

   $$\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),\left(\begin{array}{cc}0& -1\\ -1& 0\end{array}\right)\mathrm{.}$$

4. The four signed permutation matrices with one $\pm 1$ and one $\mp 1$:

   $$\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right),\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\mathrm{.}$$

Step 4: Combine with the Scalar $b$

For each of the 8 possible $2\times 2$ matrices $A$, we have two choices for $b$ ($b=1$ or $b=-1$). This gives a total of:

$$8\text{ (for }A)\times 2\text{ (for }b)=16\text{ possible matrices }Q\mathrm{.}$$

Step 5: Verify Invertibility

All orthogonal matrices are invertible by definition, so all 16 matrices $Q$ are invertible.

Conclusion

The number of $3\times 3$ invertible integer matrices $Q$ satisfying the given conditions is 16.

Final Answer: $\boxed{{\displaystyle C}}$