I have a question regarding when a matrix is diagonalizable
Suppose we have a $n \times n$ matrix $A$. Let's say that I row reduce and I find the matrix $B$, which is the row echelon form of the initial matrix $A$. If $B$ has zero rows only in the end, meaning it is of the following form
$$ \begin{bmatrix}a11 & a12 & a13 & \dots\\0 & a22 & a23 & \dots\\ 0 & 0 & a33 & \dots \\\vdots \\ 0 & 0 & 0 & \dots\\ 0 & 0 & 0 & \dots\end{bmatrix} $$
can we say that matrix $A$ is diagonalizable? If yes, then I want to ask another question. Suppose that I row reduce matrix $A$ and zero rows occur in "random" positions in the matrix(meaning that zero rows are not in the end). Is it possible to create another matrix $C$ with the same rows with $A$ but in different order, such that $C$ has the appropriate row echelon form and, hence, for it to be diagonalizable?