A $3 \times 3$ real symmetric matrix has eigenvalues $\lambda_1 = 17$, $\lambda_2 = 18$, and $\lambda_3 = 19$. The trace and determinant of the matrix are:
Options:
(A) Trace = 54, Determinant = 5814
(B) Trace = 5814, Determinant = 54
(C) Trace = 56, Determinant = 5814
(D) Trace = 54, Determinant = 5818
Answer Key:
A
Step-by-Step Solution & Explanation
By the spectral properties of square matrices:
1. The trace is the sum of the eigenvalues:
$\operatorname{tr}(A) = \sum_{i=1}^3 \lambda_i = 17 + 18 + 19 = 54$
2. The determinant is the product of the eigenvalues:
$\det(A) = \prod_{i=1}^3 \lambda_i = 17 \times 18 \times 19 = 5814$
Therefore, Trace = 54 and Determinant = 5814.