$B\in\mathbb{R}^{n\times n}$ has $n$ different rows and $\operatorname{rank} B = n-1$. Is any selection of $n-1$ rows of $B$ linearly independent?

Question

Let $B\in\mathbb{R}^{n\times n}$ a square matrix with $\operatorname{rank} B=n-1$ and $n$ different rows with at least one non-zero entry. Is it true, that any selection of $n-1$ rows of $B$ is linearly independent?

Any help is appreciated! Thank you in advance!

Answer 1

The matrix $$ \begin{bmatrix} 1 & 2 & 3 & 10 \\ 4 & 5 & 6 & 10 \\ 7 & 8 & 9 & 10 \\ 1 & 1 & 1 & 1 \end{bmatrix} $$ has rank 3. But omit the last row; the remaining rows are linearly dependent (row $2$ is the average of rows $1$ and $3$).