Eigenvalues of a matrix 1 - 1 0, 3

1. Write the characteristic equation:
\det(A - \lambda I) = 0
where:
A - \lambda I = \begin{pmatrix}1 - \lambda & -1 \\ 0 & 3 - \lambda \end{pmatrix}

2. Calculate the determinant:
\det(A - \lambda I) = (1 - \lambda)(3 - \lambda) - (-1)(0)
= (1 - \lambda)(3 - \lambda)

3. Set the determinant to zero and solve for \( \lambda \):
(1 - \lambda)(3 - \lambda) = 0
\lambda = 1 \quad \text{or} \quad \lambda = 3

Hence, the eigenvalues are:
\lambda_1 = 1 \quad \text{and} \quad \lambda_2 = 3