control systems - What is $\det[sI-A]$?

I am reading the 12th edition of "Modern Control Systems" from Richard C. Dorf and Robert H. Bishop which is about control systems.

It is not totally clear to me how the author came up with the formula in the problem's solution I describe below.

The problem: The solution: I read the theory from the corresponding chapter, but I don't totally understand how they come up with $\det[sI-A]$. I found in the theory that $[sI-A]^{-1}= \Phi$ where $\Phi$ is the matrix exponential function that describes the unforced response of the system. Yet here they modify it slightly and compute the determinant rather than the inverse. Why? How did they come up with this solution?

• Might be worth noting that $\mathrm{det}[sI - A] = 0$ gives the eigenvalues of the matrix A. – Adam Haun Feb 26 '18 at 21:41

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.
• That does clarify quite a lot! Could you just explain where $sI-A$ comes from? I'll then accept your answer – LandonZeKepitelOfGreytBritn Nov 7 '17 at 17:17
• (I only have the 11th edition of Dorf and Bishop, but I assume the contents are not that different). Section 3.7 'The time response and the state transition matrix' shows how to arrive at the response of the system, which is $(s I -A)^{-1} x(0)$. And this link (lpsa.swarthmore.edu/Transient/TransMethSS.html) also shows the derivation and works out an example similar to the one in your question. – Suba Thomas Nov 7 '17 at 17:44