change some wording of above solution ; Now compute ( \Phi(t) \begin{pmatrix}
1.25 \
0.75
\end{pmatrix} ):

Certainly! Here is the revised wording for the specified part:

Now calculate \(\Phi(t) \begin{pmatrix}
1.25 \\
0.75
\end{pmatrix}\):

\[
\begin{pmatrix}
e^{4t} & 1 \\
-2e^{4t} & 2
\end{pmatrix} \begin{pmatrix}
1.25 \\
0.75
\end{pmatrix} = \begin{pmatrix}
e^{4t}(1.25) + 0.75 \\
-2e^{4t}(1.25) + 2(0.75)
\end{pmatrix} = \begin{pmatrix}
1.25e^{4t} + 0.75 \\
-2.5e^{4t} + 1.5
\end{pmatrix}
\]

Is there anything else you would like to clarify or any further details you need? Here are five more questions you can ask:

1. How do you find the eigenvalues and eigenvectors of a matrix?
2. What is the importance of the fundamental matrix in solving differential equations?
3. Can you explain the process of finding the inverse of a matrix?
4. How do you interpret the solution of a system of differential equations?
5. What are some applications of solving systems of differential equations in real-world scenarios?

**Tip:** When solving differential equations, always verify the consistency of your initial conditions with the solution to ensure correctness.

Learn how to compute \(\Phi(t)\) for a matrix multiplication involving exponential functions. This problem involves matrix-vector multiplication and exponential function applications. Suitable for advanced mathematics learners.

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