Discussion
The eigenvalues of the matrix \(\begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}\) are _____.
Answer: C
The eigenvalues are 1 and 3.
The matrix is upper triangular, and the eigenvalues of a triangular matrix are its diagonal entries. Formally:
\[ \det\begin{bmatrix} 1 - \lambda & 2 \\ 0 & 3 - \lambda \end{bmatrix} = (1 - \lambda)(3 - \lambda) = 0 \]
Check: the eigenvalues sum to the trace, 1 + 3 = 4, and multiply to the determinant, 3. The pairs (0, 2), (2, 3) and (1, 2) fail one or both of these checks; the off-diagonal 2 does not affect the eigenvalues of a triangular matrix.
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