The problem. Find a fundamental matrix and solve for \(\displaystyle\Phi{\left({0}\right)}={I}\). First determine the eigenvalues and their corresponding eigenvectors. After that, determine a general solution of the d.e Then construct a fundamental matrix. After tha, solve the constants \(\displaystyle{c}_{{1}},{c}_{{2}},{c}_{{3}},{c}_{{4}}\) for the two conditions following from

\[\Phi(0)=\begin{bmatrix}1&0\\0&1\end{bmatrix}\]

\[x'=\begin{bmatrix}-1&-4\\1&-1\end{bmatrix}\]

Determine the eigenvalues

\(\displaystyle{\left(-{1}-\lambda\right)}^{{2}}+{4}={0}\to\lambda^{{2}}+{2}\lambda+{5}={0}\to\lambda=-{1}\pm{2}{i}\)

Row-reduce \(A-\lambda I\) to echelon and the eigenvectors follow.

\[v_1=\begin{bmatrix}2i\\1\end{bmatrix}\]

\[v_2=\begin{bmatrix}-2i\\1\end{bmatrix}\]

Consider a general solution in the form of a fundamental matrix.

\[x=\begin{bmatrix}x_1&x_2\end{bmatrix}c=e^{-t}\begin{bmatrix}-2\sin(2t)&2\cos(2t)\\\cos(2t)&\sin(2t)\end{bmatrix}c\]

Solve the system of equations that follow from

\[\Phi(0)=\begin{bmatrix}1&0\\0&1\end{bmatrix}\]

\[\Phi(0)=\begin{bmatrix}1&0\\0&1\end{bmatrix}\to\]

\(\displaystyle{c}_{{1}}={0}\)

\(\displaystyle{2}{c}_{{2}}={1}\)

\(\displaystyle{c}_{{3}}={1}\)

\(\displaystyle{c}_{{4}}={0}\)

Substitute the c-values and get the final result.

\[\Phi(t)=e^{-t}\begin{bmatrix}\cos(2t)&-2\sin(2t)\\\frac{1}{2}\sin(2t)&\cos(2t)\end{bmatrix}\]