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Consider the matrix P = [ 2 0 0 ] [ 0 2 0 ] [ 0 0 3 ]. Let the transpose of a matrix X be denoted by Xᵀ. Then the number of 3 × 3 invertible matrices Q with integer entries, such that Q⁻¹ = Qᵀ and PQ = QP, is

1. 32
2. 8
3. 16
4. 24

Correct answer: 16

Solution

For Q⁻¹ = Qᵀ, the matrix Q must be orthogonal, meaning QᵀQ = I. Additionally, the condition PQ = QP implies Q must commute with P, which restricts Q to a specific form where it preserves the eigenvalues of P. Considering these constraints and the fact that Q has integer entries, there are exactly 16 such matrices that satisfy both conditions.

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