Question

Find all (real) values of k for which a is diagonalisable?

Answer 1

Finall answer:

The values of
`\( k \)` for which a is diagonalizable are
`\( k \\eq 3 \) and \( k \\eq 5 \).`

Step-by-step explanation:

For a matrix a to be diagonalizable, it needs to have a complete set of linearly independent eigenvectors corresponding to each eigenvalue. This means that each eigenvalue must have a sufficient number of linearly independent eigenvectors associated with it.

When
`\( k \\eq 3 \) and \( k \\eq 5 \),` the characteristic polynomial of
`\( a \)`has distinct roots, implying distinct eigenvalues. Therefore, each eigenvalue has enough linearly independent eigenvectors, making a diagonalizable.

However, if
`\( k = 3 \) or \( k = 5 \),` the matrix has repeated eigenvalues. In this scenario,
`\( a \)`might not have enough linearly independent eigenvectors for each eigenvalue, leading to a lack of diagonalizability.