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Let P and Q be two square invertible matrices such that Q = -P^(-1) * Q * P. What is (P + Q)² equal to?
- Null matrix
- P² + 2*P*Q + Q²
- (P - Q)²
- Identity matrix
Correct answer: (P - Q)²
Solution
From Q = -P^(-1)*Q*P, multiply both sides on the left by P: P*Q = -Q*P, so P*Q + Q*P = 0. Then (P+Q)² = P² + P*Q + Q*P + Q² = P² + 0 + Q² = P² + Q². Also (P-Q)² = P² - P*Q - Q*P + Q² = P² + 0 + Q² = P² + Q². So (P+Q)² = (P-Q)².
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