Find the rank of the matrix $A=\begin{pmatrix} 2 & 3 & -1 & 1\\ 3 & 0 & 4 & 2\\ 6 & 9 & -3 & 3 \end{pmatrix}$.
Now, I know that the rank of a matrix is actually equal to the greatest positive integer $r$ such that the matrix has atleast a non-zero minor of order $r$. However, I want to know that how to calculate the minor of an element of a non-square, i.e., a 'purely' rectangular matrix. I know how to calculate the minors of a square matrix of each "elements" , but I don't get how to calculate minors of order $k$, in general, from a given rectangular matrix? Like in the above example how to calculate minors of order $3$?
The determinant of any of the four $3\times3$ submatrices of $A$ is $0$, and therefore the rank is at most $2$. But $\left|\begin{smallmatrix}2&3\\3&0\end{smallmatrix}\right|=-9\ne0$, and so $\operatorname{rank}(A)=2$.