Let $A$ be an $n \times n$ upper triangular matrix with complex entries. Pick out the true statement(s):
$(a)$ If $A \neq 0$, and if $a_{ii} = 0$, for all $1\leq i \leq n$, then $A^n = 0$.
$(b)$ If $A \neq I$ and if $a_{ii} = 1$ for all $1\leq i \leq n$, then $A$ is not diagonalizable.
$(c)$ If $A \neq 0$, then $A$ is invertible.
$(a)$ is true as eigenvalues of the upper triangular matrices are diagonal elements and here all the eigenvalues are $0$. Hence $x^n=0$ is the characteristic equation and by Cayley-Hamilton theorem $A^n=0$.
I have no idea about $(b)$.
$(c)$ is not true.