For any real symmetric matrix $A$, which of the following statements is/are always TRUE?
Options:
(A) All eigenvalues of $A$ are real numbers
(B) Eigenvectors corresponding to distinct eigenvalues are mutually orthogonal
(C) $A$ is always orthogonally diagonalizable
(D) The determinant of $A$ is equal to the product of its eigenvalues
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
By the Spectral Theorem for real symmetric matrices:
1. All eigenvalues of a real symmetric matrix are strictly real (A is true).
2. If $Ax_1 = \lambda_1 x_1$ and $Ax_2 = \lambda_2 x_2$ with $\lambda_1 \ne \lambda_2$, then $(\lambda_1 - \lambda_2)(x_1^T x_2) = 0 \implies x_1^T x_2 = 0$, meaning eigenvectors are orthogonal (B is true).
3. There exists an orthogonal matrix $Q$ such that $Q^T A Q = D$, where $D$ is diagonal (C is true).
4. The determinant of any square matrix is the product of its eigenvalues: $\det(A) = \prod \lambda_i$ (D is true).