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Let P and Q be two 2x2 real matrices such that PQ equals the null matrix and trace(P) = trace(Q) = 0. Which of the following must be true?

1. P and Q commute under matrix multiplication
2. P and Q do not commute under matrix multiplication
3. P² Q is not a null matrix
4. None of these

Correct answer: P and Q commute under matrix multiplication

Solution

Since tr(P) = tr(Q) = 0, both matrices satisfy A² = (det A)I. From PQ = O, the product of determinants is zero. Using these constraints one can show QP = O as well, so PQ = QP = O, i.e., P and Q commute.

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