If $A$ is a $3 \times 3$ real orthogonal matrix with $\det(A) = 1$, which of the following is always true?

(A) $A^T = A^{-1}$

(B) $A = A^T$

(C) All eigenvalues of $A$ are real and equal to 1

(D) $\text{Trace}(A) = 3$

Answer Key:

A

### Solution: By definition of an orthogonal matrix, $A^T A = I$, which implies $A^T = A^{-1}$. Correct: A.

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