Discussion
The eigenvalues of the matrix \(\begin{bmatrix} 6 & 1 \\ -2 & 3 \end{bmatrix}\) are
Answer: C
The eigenvalues are 5 and 4.
Solve the characteristic equation \(\det(A - \lambda I) = 0\):
\[ \begin{aligned} (6 - \lambda)(3 - \lambda) - (1)(-2) &= 0 \\ \lambda^{2} - 9\lambda + 20 &= 0 \\ (\lambda - 5)(\lambda - 4) &= 0 \end{aligned} \]
A quick check: the eigenvalues must add to the trace, 6 + 3 = 9, and multiply to the determinant, 18 + 2 = 20. The pair (3, 6) just reads off the diagonal, which works only for a triangular matrix; (1, −2) and (1, 6) fail both the sum and the product test.
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