Let \(A\) be a \(2 \times 2\) matrix: \[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \]
The characteristic polynomial \(p(\lambda)\) is given by: \[ p(\lambda) = \det(A - \lambda I) \]
Substitute the values into the determinant: \[ p(\lambda) = \det\left(\begin{bmatrix} 1 - \lambda & 2 \\ 3 & 4 - \lambda \end{bmatrix}\right) \]
Expand the determinant: \[ p(\lambda) = (1 - \lambda)(4 - \lambda) - 2*3 \]
\[ p(\lambda) = \lambda^2 - \lambda - 4\lambda + 4 - 6 \]
\[ p(\lambda) = \lambda^2 - 5\lambda - 2 \]