Rank of a matrix if one of the diagonal elements is $0$ during elementary row operations

Question

I'm required to find the rank of a square matrix A of dimension $m * m$. While doing elementary row operations I encountered a $0$ at a diagonal position. I tried all the possible row exchanges, but none of these could get a nonzero number at that location.

At this step what can one tell about the rank of the matrix? Is it $0$ or $m-1$ or something else? And what is the reasoning behind it?

Answer 1

Let $A,B \in \mathcal{M}_{3 \times 3}(\mathbb{K})$ be defined as

\begin{align*} A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix} && B = I_3 \end{align*}

Matrix $A$ is obtained from $B$ by performing row operations, but yet $\text{rank}(A) = \text{rank}(B) = 3$.

But there is something of interest. If the question was

What does it mean for the rank of a matrix if we obtain a zero at the diagonal in the row reduced form?

And from this you can conclude that (for example) if $A$ is an $n\times n$ matrix and $m$ is the number of zeros that you find in this situation, then $\text{rank}(A) = n -m$.