Chapter 9.3, Problem 39E

To determine

(a)

To find:

The value of the ${\displaystyle \int \text{A}\left(t\right)dt}$ for the matrix $\text{A}\left(t\right)=\left[\begin{array}{cc}t& e^t\\ 1& e^t\end{array}\right]$.

To determine

(b)

To find:

The value of the ${\displaystyle {\int }_0^1\text{B}\left(t\right)dt}$ for the matrix $\text{B}\left(t\right)=\left[\begin{array}{cc}\cos t& -\sin t\\ \sin t& \cos t\end{array}\right]$.

To determine

(c)

To find:

The value of $\frac{d}{dx}\left[\text{A}\left(t\right)\text{B}\left(t\right)\right]$ for the given matrix.