check the validity of the statement

Question

Let $A$ ne an $n\times n $ upper triangular matrix with complex enries.I have to check the validity of the statement

If $A\neq I$ and if $a_{ii}=1$ $\forall i, 1\le i\le n$, then $A$ is not diagonalizable.

I wanted to know whether my proof(given below) is correct or not?Please provide the correct proof...

Answer 1

The claim that there is no pivot volumn is false.

Consider an explicit example: $A = \begin{bmatrix}1 & 1 \\ 0 & 1 \end{bmatrix}$

then we have $A-I=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$

The second column is a pivot column.

In general, we know that for your setting, we are going to have positive number of pivot columns.