I’m looking into MATLAB’s state-space functionality, and I found a peculiar relation that I don’t believe I’ve seen before, and I’m curious how one might obtain it. According to this documentation page, when converting a state-space system representation to its transfer function, the following well-known equality is used
$H(s) = C(sI-A)^{-1}B$
However, they go one step further and state that
$H(s) = C(sI-A)^{-1}B = \frac{{\det}(sI-A+BC) - {\det}(sI-A)}{\det(sI-A)}$
How did they manage to convert $C{\text {Adj}}(sI-A)B$ into ${\det}(sI-A+BC) - {\det}(sI-A)$?
As far as I understand, we can assume here that $B$ is a column vector and $C$ is a row vector, i.e. it’s a single-input / single-output relationship.