Q:

Rank of matrix $M$

let $M$ a matrix over $M(m,k,\mathbb{K})$ and matrix $B$ over $M(m,l,\mathbb{K})$, wath is the sufficient condition for $\operatorname{rank}(\lbrack M\mid B \rbrack ) = \operatorname{rank}(M)$?

A:

(Assuming I've understood your notation correctly)
You need each column of $B$ to be a linear combination of columns of $M$.

