Rank of a matrix. Is it correct?

Question

Given the matrix $B∈ M_3(Z_7)$

$$ \begin{bmatrix} 2 & 1 & 0 \\ 1 & 3 & 1 \\ 4 & 0 & 2 \\ \end{bmatrix} $$

Is the rank of this matrix $2$? I calculated the determinant of the matrix

$$ \begin{bmatrix} 2 & 1 \\ 1 & 3 \\ \end{bmatrix} $$

and it is equal to $5$ so since the determinant of this $2 x 2$ sub-matrix is non-zero it means that the rank is $2$, right?

Answer 1

You are right if $\Delta=0$ and indeed, $\Delta=0.$

Answer 2

Note that the rank of $B$ can be viewed as $n$ where $n$ is the size of the largest non-zero $n\times n$ sbmatrix with non-zero determinant

In this case $\det B=12$ then $\text{rank} (B)=3$.

Rank