Chapter 9.3, Problem 40E

To determine

(a)

To find:

The value of the ${\displaystyle \int \text{A}\left(t\right)dt}$ for the matrix $\text{A}\left(t\right)=\left[\begin{array}{cc}1& e^{-2t}\\ 3& e^{-2t}\end{array}\right]$

To determine

(b)

To find:

The value of the ${\displaystyle {\int }_0^1\text{B}\left(t\right)dt}$ for the matrix $\text{B}\left(t\right)=\left[\begin{array}{cc}{e}^{-t}& e^{-t}\\ -e^{-t}& 3e^{-t}\end{array}\right]$.

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.