Let \(B\) be a \(2 \times 2\) matrix:
\[ B = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} \]
The characteristic polynomial \[p(\lambda)\] is given by: \[ p(\lambda) = \det(B - \lambda I) \]
Now calculate \[(λI)\] where I is the identity matrix
\[ \lambda I = \begin{bmatrix} 2-\lambda & -1 \\ 1 & 1-\lambda \end{bmatrix} \]
Substitute the values into the determinant:
\[ p(\lambda) = \det\left(\begin{bmatrix} 2-\lambda & -1 \\ 1 & 1-\lambda \end{bmatrix}\right) \]
Expand the determinant:
\[ p(\lambda) = (2 - \lambda)(1 - \lambda) + 1\]
which gives the roots of this polynomial equation which are the eigenvalues
\[ \lambda = 1 \] \[ \lambda = -3 \]