Find the characteristic polynomial of the matrix
\(A=\left[ \begin{array}{c} 1 & 2 \\ 3 & 4 \end{array} \right]\)
The first step taken was to subtract the identity matrix multiple from the matrix \(A\)
\[ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} - \begin{bmatrix} \lambda & 0 \\ 0 & \lambda \end{bmatrix} \]
Using this formula \(A - \lambda I\) we calculate the following
\(A=\left[ \begin{array}{c} 1-\lambda & 2 \\ 3 & 4-\lambda \end{array} \right]\)
Next we calculate the determinant of the matrix
\((1-\lambda)(4-\lambda)-(2)(3)\)
\(=4 - \lambda - 4\lambda + \lambda^{2} - 6\)
\(=4 - 5\lambda + \lambda^{2} -6\)
Leaving us with the characteristic polynomial \(\lambda^{2} - 5\lambda - 2=0\)