\[\mathbf{A} = \left[\begin{array} {rrr} 1 & 2 \\ 3 & 4 \\ \end{array}\right] \]
\[\mathbf{det(A - xI)} =det(\left[\begin{array} {rrr} 1 & 2 \\ 3 & 4 \\ \end{array}\right] - x \left[\begin{array} {rrr} 1 & 0 \\ 0 & 1 \\ \end{array}\right] ) \] \[\mathbf{det(A - xI)} =det(\left[\begin{array} {rrr} 1-x & 2 \\ 3 & 4-x \\ \end{array}\right]) \]
\[det(A - xI) = x^2 - 5x - 2\]