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This image contains a set of linear algebra and matrix-related questions. I'll provide a summary of each question to assist you further. Let me know if you'd like me to go through specific problems, or if you want detailed solutions for any or all questions.

### Problem Summaries:

1. **Q1**: Test the consistency and solve the system of linear equations given.
2. **Q2**: Determine values of \( \lambda \) and \( \mu \) for the consistency of another system of linear equations with conditions for no solution, unique solution, and infinite solutions.
3. **Q3**: Find the eigenvalues and eigenvectors (characteristic roots and vectors) for two matrices.
4. **Q4**: Verify the Cayley-Hamilton theorem for a given matrix.
5. **Q5**: Identify values of \( \lambda \) and \( \mu \) that ensure the given system has no solution, a unique solution, or infinitely many solutions.
6. **Q6**: Similar to Q2, investigate the values of \( \lambda \) and \( \mu \) for consistency types in a system of linear equations.
7. **Q7**: Show the consistency of a system of equations and solve it if consistent.
8. **Q8**: Determine the rank and nullity of two given matrices.
9. **Q9**: Find the normal form of a matrix and calculate its rank.
10. **Q10**: Apply elementary transformations to a given matrix.
11. **Q11**: Prove that a specific matrix is not diagonalizable over the field of complex numbers.
12. **Q12**: Diagonalize a given matrix.
13. **Q13**: Use the Cayley-Hamilton theorem to express a matrix expression as a polynomial of the matrix \( A \).

Please let me know which specific problem(s) you'd like help with, or if you’d like solutions to all of them sequentially. Here’s how I can assist further. 

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#### Follow-Up Questions for Expanding Knowledge:

1. How does one determine the consistency of a linear system using augmented matrices?
2. What are the steps to find the eigenvalues and eigenvectors of a matrix?
3. What is the significance of the Cayley-Hamilton theorem in linear algebra?
4. How do elementary transformations affect the rank of a matrix?
5. Why are some matrices not diagonalizable, even over complex fields?

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#### Tip:
For any system of linear equations, using the row echelon form can simplify the process of checking for consistency and finding solutions.

A set of linear algebra and matrix-related questions covering topics like consistency of linear equations, eigenvalues and eigenvectors, Cayley-Hamilton theorem, matrix rank, and diagonalization.

Explore solutions to various linear algebra problems including checking consistency of linear systems, finding eigenvalues and eigenvectors, verifying the Cayley-Hamilton theorem, calculating matrix rank and nullity, and diagonalizing matrices. Suitable for college-level mathematics students.

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