Q4: Given the matrix
Where
Find:

1. the rank of ?
2. the eigen values and eigen vectors?
3. satisfy or ?

To start with the first question, the rank of the matrix can be computed through methods like Gaussian elimination or finding its row echelon form. In this case, the rank of is 3, indicating that the matrix has three linearly independent rows or columns—how eloquent is that?

Now, moving onto the eigenvalues and eigenvectors! By computing the characteristic polynomial (determinant of ), you will identify the eigenvalues, which turn out to be . The corresponding eigenvectors can be found by solving for each eigenvalue. Voila! You have your treasures of eigenvalues and eigenvectors!

Lastly, regarding the matrix relation , this representation can indeed be achieved if the columns of are made up of the normalized eigenvectors of . This settles our query with a ‘yes,’ presenting a delightful result of diagonal dominance in understanding the system!