Characteristic polynomial of $$$\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right]$$$

Find the characteristic polynomial of $$$\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right]$$$.

Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix:

$$$\left[\begin{array}{cc}1 - \lambda & 2\\3 & 4 - \lambda\end{array}\right]$$$

The characteristic polynomial is the determinant of the obtained matrix:

$$$\left|\begin{array}{cc}1 - \lambda & 2\\3 & 4 - \lambda\end{array}\right| = \lambda^{2} - 5 \lambda - 2$$$ (for steps, see determinant calculator).

The characteristic polynomial is $$$p{\left(\lambda \right)} = \lambda^{2} - 5 \lambda - 2$$$A.