Hello, could someone help me with these linear algebra assignment questions?

It is on eigenvalue and eigen vectors.

If there is anything that that might be needed, I can try going through the notes, please let me know.

(4. Suppose that T : V V is a linear transformation and 1 is finite dimensional. Prove that T is an isomorphism if and only if no eigenvalue of T is equal to (0. (5. We checked carefully that det is invariant under the row operations. In this question. vou will show that eigenvalues are (unfortunately) not invariant. R.+R (a) Give an example of a matrix M such that applying a row operation Af SR At produces new eigenvalues. [Your example should have explicit numerical entries.) {b) Prove that the eigenvalue A = 0 is always preserved under row operations. (6. Suppose that M = [i 2] satisfies a = d and b = c. {a) Prove that M has an eigenvalue with algebraic multiplicity 2 if and only if b= 0. {b) Suppose A; and A; are the eigenvalues of M. If Ay = As, what can be said about a? (} Compute the eigenvalues of M in terms of @ and b. (37. Let A be a real n by n matrix with a complex eigenvalue. Show that any eigenvector that corresponds to a complex eigenvalue must have an entry with at least one entry that is complex