If $A$ is an orthogonal $3 \times 3$ matrix with real entries, which of the following statements is ALWAYS TRUE?

(A) \det(A) = \pm 1

(B) \det(A) = 0

(C) A = A^T

(D) \text{trace}(A) = 3

Answer Key:

A

### Solution: For an orthogonal matrix, $A^T A = I \implies \det(A^T)\det(A) = (\det(A))^2 = 1 \implies \det(A) = \pm 1$. Correct: A.

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