Show that the discrete$-$time state$-$space system

$x(k+1)=Ax(k)+Bu(k)$

$y(k)=Cx(k)+Du(k)$

has impulse response

$g(k)=\{\begin{array}{ccc}D& if& k=0\\ C{A}^{k-1}B& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

HINT: Brute$-$force solve the dynamics with the zero initial condition $x(0)=0$ and

impulse input

$u(k)=\{\begin{array}{ccc}1& if& k=0\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$