control systems - What is \$\det[sI-A]\$?
The inverse is \$(s I -A)^{-1}= \frac{1}{\det (s I -A)} \text{adj} (s I -A)\$ .

The numerator which is the adjoint of \$s I -A\$ contributes to the zeros of the system.

The denominator which is the determinant of \$s I -A\$ contributes to the poles of the system.

Only the poles affect stability. So if you are only interested in the stability of the system, just compute the determinant.

To get the system's response you have to compute the adjoint as well.