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The image you've uploaded contains a math exam or worksheet focused on linear algebra, specifically matrix operations. To address the problems and explain solutions, I'll need to transcribe the questions. Let me extract the information from the image and analyze it.

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**Questions:**

1. Prove that for all \( A \in M_2(\mathbb{R}) \), \( A^2 = 0 \) (null matrix) implies that \( A \) is similar to the matrix:
   \[
   \begin{pmatrix}
   0 & 1 \\
   0 & 0
   \end{pmatrix}
   \]
   Show the condition under which the operator \( A \) is not diagonalizable.

2. Find the matrix of a linear operator \( L \) with respect to the basis \( (e_1, e_2) \), given:
   \[
   L(e_1) = e_2, \quad L(e_2) = 0
   \]

3. Let \( A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 1 & 0 & 1 \end{pmatrix} \), find \( A^n \) for any \( n \in \mathbb{N} \).

4. Give an example of two \( 2 \times 2 \) real matrices \( A \) and \( B \) such that \( AB = BA \), but \( A \neq B \), and neither \( A \) nor \( B \) is a scalar matrix.

5. Let \( A \in M_2(\mathbb{R}) \), show that if \( A \) is diagonalizable, then there exists \( P \in GL_2(\mathbb{R}) \) such that \( P^{-1} A P = D \), where \( D \) is a diagonal matrix. Compute \( A^n \) in this case.

6. Let \( A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \) and \( B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \). Show that \( A B = BA \).

7. Let \( A \in M_n(\mathbb{R}) \) such that \( A^2 = A \), find all eigenvalues of \( A \).

8. Compute the matrix \( A^n \) for the following matrix \( A \):
   \[
   A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}
   \]

9. If \( A \in M_2(\mathbb{R}) \), show that if \( A^2 = 0 \), then \( A \) is similar to the matrix \( \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \).

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Would you like me to start with the solution for a particular problem? Or shall I go through them in order?

Here are 5 relative questions to expand the discussion:

1. How does matrix diagonalization relate to similar matrices?
2. What does it mean for a matrix to be idempotent (\( A^2 = A \)) in terms of its eigenvalues?
3. How can you determine if a matrix is diagonalizable from its Jordan form?
4. What is the significance of a nilpotent matrix, where \( A^n = 0 \)?
5. How can you compute matrix powers efficiently, especially for triangular or block matrices?

**Tip:** When working with matrix powers, it's useful to know their eigenvalues since \( A^n \) behaves simply if you diagonalize the matrix.

Algebra 3 Matrix Series Exam Questions including matrix operations, diagonalization, eigenvalues, and matrix powers.

Explore advanced matrix problems including diagonalization, eigenvalue computation, and nilpotent matrices. Learn how to solve for matrix powers and apply the Jordan canonical form. Suitable for university-level linear algebra courses.

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