Suppose the $8 \times 4$ matrix $A$ has rank $4$. Is it always true that any $4 \times 4$ submatrix of $A$ has rank $4$? I am doing research on coding theory and I am wondering whether this is true.
My guess is that it is always true. Since $A$ has rank $4$, any $4$ rows are linearly independent.
Remark: I am considering matrix of the form
![enter image description here]
where $\alpha, \beta, \gamma$ are non-zero. In this case, my question is: any $3\times 3$ submatrix of the matrix above has rank $3$. Is the statement true? Note that the matrix above is assumed to have rank $4$.