Let $A$ be a $3 \times 3$ real symmetric matrix with eigenvalues $\lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 3$. Which of the following statements are necessarily TRUE?
Options:
(A) The determinant of matrix $A$ is equal to 6
(B) The trace of matrix $A$ is equal to 6
(C) The eigenvectors corresponding to distinct eigenvalues of $A$ are mutually orthogonal
(D) Matrix $A$ is positive definite
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
1. The determinant of a square matrix is the product of its eigenvalues: $\det(A) = \lambda_1 \lambda_2 \lambda_3 = 1 \times 2 \times 3 = 6$. (Statement A is True)
2. The trace of a square matrix is the sum of its eigenvalues: $\text{Tr}(A) = 1 + 2 + 3 = 6$. (Statement B is True)
3. For any real symmetric matrix, eigenvectors corresponding to distinct eigenvalues are mutually orthogonal. (Statement C is True)
4. Since all eigenvalues of the real symmetric matrix are strictly positive ($\lambda_i > 0$), $A$ is positive definite. (Statement D is True)