Which of the following statements regarding the eigenvalues and eigenvectors of a real symmetric matrix $A$ are CORRECT?
Options:
(A) All eigenvalues of a real symmetric matrix are real numbers.
(B) Eigenvectors corresponding to distinct eigenvalues are mutually orthogonal.
(C) The trace of $A$ equals the sum of its eigenvalues.
(D) The determinant of $A$ equals zero for all real symmetric matrices.
Answer Key:
A, B, C
Step-by-Step Solution & Explanation
Statements A, B, and C are fundamental spectral theorems for real symmetric matrices: eigenvalues are all real, eigenvectors from distinct eigenspaces are orthogonal, and the sum of eigenvalues equals trace. Statement D is false because real symmetric matrices can be non-singular with non-zero determinants.