Answer
See below
Work Step by Step
According to Cramer's Rule, the solution to $2 \times 2$ system $Ax=b$ is given by:
$$x_k=\frac{\det(B_k)}{\det(A)}$$
From the given matrix, we have:
$A=\begin{bmatrix}
e^t & e^{-2t} \\e^t & -2e^{-2t}
\end{bmatrix}\rightarrow det(A)=-2e^{-t}-e^{-t}=-3e^{-t}\\
B=\begin{bmatrix}
3\sin t& e^{-2t} \\4\cos t & -2e^{-2t}
\end{bmatrix} \rightarrow \det(B)=-2e^{-2t}(3\sin t+2\cos t)\\
C=\begin{bmatrix}
e^t & 3\sin t \\e^t & 4\cos t
\end{bmatrix} \rightarrow \det(C)=e^t(4\cos t-3\sin t)$
Hence, $x_B=\frac{-2e^{-2t}(3\sin t+2\cos t)}{-3e^{-t}}\\
x_C=\frac{e^t(4\cos t-3\sin t)}{-3e^{-t}}$
