A $3 \times 3$ real symmetric matrix has eigenvalues $\lambda_1 = 5$, $\lambda_2 = 6$, and $\lambda_3 = 7$. The trace and determinant of the matrix are:
Options:
(A) Trace = 18, Determinant = 210
(B) Trace = 210, Determinant = 18
(C) Trace = 20, Determinant = 210
(D) Trace = 18, Determinant = 214
Answer Key:
A
Step-by-Step Solution & Explanation
By the spectral properties of square matrices:
1. The trace is the sum of the eigenvalues:
$\operatorname{tr}(A) = \sum_{i=1}^3 \lambda_i = 5 + 6 + 7 = 18$
2. The determinant is the product of the eigenvalues:
$\det(A) = \prod_{i=1}^3 \lambda_i = 5 \times 6 \times 7 = 210$
Therefore, Trace = 18 and Determinant = 210.