Chapter 4.7, Problem 1P

(a) If

$X\left(t\right)=\left(\begin{array}{cc}{x}_{11}\left(t\right)& x_{12}\left(t\right)\\ x_{21}\left(t\right)& x_{22}\left(t\right)\end{array}\right)$ and $u\left(t\right)=\left(\begin{array}{l}{u}_1\left(t\right)\\ u_2\left(t\right)\end{array}\right)$,

show that ${\left(Xu\right)}^{\prime }=X^{\prime }u+X{u}^{\prime }$.

(b) Assuming that $X\left(t\right)$ is a fundamental matrix for $x\text{'}=P\left(t\right)x$ and that $u\left(t\right)={\displaystyle \int X^{-1}\left(t\right)g\left(t\right)dt}$, use the result of part (a) to verify that $x_p(t)$ given by Eq. $\left(15\right)$ satisfies Eq. $\left(1\right)$, $x\text{'}=P\left(t\right)x+g\left(t\right)$.