P$6\mathrm{.}21$ Consider the system described in state variable form by

$($t$)=$ Ax $($t$)+$ Bu $($t$)$

y $($t$)=$ Cx $($t$)$

where

A $=$

K$1$

$1$

$$k$2$

B $=$

$1$

~

and C $=[1-1],$

and where k$1$ kz and both k$1$ and kz are real numbers.

a$.$ Compute the state transition matrix @ $($t$,0).$

b$.$ Compute the eigenvalues of the system matrix A$.$

c$.$ Compute the roots of the characteristic polynomial. $($d$)$

Discuss the results of parts $($a$)-($c$)$ in terms of stability of the system.