Let $V$ be a finite dimensional vector space over $\Bbb R$ and let $T :V \to V$ be a linear transformation. Let $A$ be the matrix of $T$ with respect to the standard basis for $V$. For each of the following assertions, state whether it is true or not. If the assertion is true, briefly justify why this is so; otherwise, provide a counter-example.
(a) If $T$ is diagonalisable then each of its eigenvalues has algebraic multiplicity equal to $1$.
(b) If none of the eigenvalues of $T$ are zero then the determinant of $A$ is not zero.
(c) If $v_1$ and $v_2$ are eigenvectors for $T$ associated to eigenvalues $\lambda_1$ and $\lambda_2$, respectively, then $v_1 + v_2$ is an eigenvector for $T$ with associated eigenvalue $\lambda_1 + \lambda_2$.
For (a) I gave a counterexample and said it was false. Is this the correct answer?
For (b) I think it is correct since the determinant of characteristic polynomial of $A$ can only be zero if there is no eigenvalue. Is this correct? If so is there a theorem that proves that?
For (c) I said it is false since eigenvalues added together does not form a new eigenvalue for $T$, is this correct?
Thanks for the help!