This document contains problems from a homework assignment in Linear Algebra. Here's an outline of the questions:

1. Initial Value Problem: Solve $\frac{du}{dt} = Au$ with the given matrix $A$ and initial condition $u(0)$.

2. Matrix Exponential: Diagonalize $A$ and compute $e^{At}$ for a given $2 \times 2$ matrix.

3. Fourth-Order ODE: Solve the fourth-order linear ordinary differential equation.

4. True/False Questions: Answer with reasoning or counterexamples for various statements about diagonalizability and properties of matrices.

Would you like help solving one or all of these problems? Let me know if you'd like details or explanations for any of the questions.

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Follow-Up Questions:

1. How would you approach solving the initial value problem using eigenvalues and eigenvectors of $A$?
2. What is the general formula for $e^{At}$ for a diagonalizable matrix $A$?
3. What techniques can simplify solving high-order linear ODEs like the one in Question 3?
4. Why might the determinant being zero affect diagonalizability in Question 4(a)?
5. Can a symmetric matrix with complex eigenvalues ever be diagonalizable over real numbers?

Tip:

When tackling matrix exponentials like $e^{At}$, diagonalizing $A$ (if possible) simplifies computation significantly, reducing it to exponentials of eigenvalues.