The uploaded document contains several advanced mathematical problems, including topics in linear algebra (eigenvalues, eigenvectors, matrix rank, and transformations), calculus, and other areas of algebra. I'll start by solving one of the questions from the document.

Let me know which problem you would like assistance with, or I can provide a detailed solution for a selected question from the list. For example, I can start by solving an eigenvalue problem, proving a matrix-related theorem, or analyzing a system of equations.

Do you have a specific question in mind from this list? Or would you like me to choose one to demonstrate a solution?

Here are some follow-up questions you might consider:

1. Would you like to see how to find eigenvalues of a matrix (e.g., Q.10)?
2. Do you need assistance with a proof involving matrices (e.g., Q.7 or Q.8)?
3. Should I solve a system of equations for consistency (e.g., Q.21)?
4. Would you prefer guidance on using elementary transformations to find a matrix rank (e.g., Q.12)?
5. Do you need help with diagonalizing matrices (e.g., Q.25)?

Tip: For eigenvalue problems, it's important to first set up the characteristic equation by subtracting $\lambda I$ from the matrix and calculating the determinant.