I have the following initial value problem (IVP):
$$ \mathbf{X}^{\prime}=\mathbf{A} \mathbf{X}, \qquad \mathbf{X}(0)=\mathbf{I} $$
$$ \text { If } \mathbf{x}(s)=\mathscr{L}\{\mathbf{X}(t)\}=\mathscr{L}\left\{e^{\mathbf{A} t}\right\} $$
$$ s \mathbf{x}(s)-\mathbf{X}(0)=\mathbf{A} \mathbf{x}(s) \quad \text { or } \quad(s \mathbf{I}-\mathbf{A}) \mathbf{x}(s)=\mathbf{I} $$
I've been a little confused with the use of Identity as opposed to a standard multiplication without the identity. How does $s \mathbf{x}(s)-\mathbf{X}(0)=\mathbf{A} \mathbf{x}(s)$ become $(s \mathbf{I}-\mathbf{A}) \mathbf{x}(s)=\mathbf{I}$?
Sorry about the simplicity here but it's always confused me.