How to derive the higher order linear ODE from autonomous $1$st order ODE?

Question

I know the following:

How to reduce higher order linear ODE to a system of first order ODE?

Now I want to consider the revsersed side:

Suppose I have the following:

$$\begin{bmatrix} \dot{y_1}\\ \dot{y_2}\\ \dot{y_3}\end{bmatrix}=\begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\end{bmatrix}\begin{bmatrix} y_1\\ y_2\\ y_3\end{bmatrix} \longrightarrow \dot{\hat{y}} = A\hat{y}$$

If $A$ is not a function of $t$, how to show the following ($i=1,2,3$):

$$\dddot{y}_i + b_2\ddot{y}_i+b_1\dot{y}_i+b_0y_i = 0$$

can be expressed as

$$\det\big(A-I(\frac{d}{dt})\big)y_i(t)=0$$

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How to derive it? Can anyone give me the sketch of the derivation or a hint? Thanks!

Answer 1

Just apply Cayley-Hamilton, $$ \chi_A(\frac d{dt})y(t)=\chi_A(A)y(t)=0. $$