How do I know if the rows of matrix are linearly independent?

Question

Given is matrix:

$G'=\begin{pmatrix} 1& 0& 1& 1& 0& 1& 0& 1& \\ 1& 1& 0& 1& 1& 1& 1& 1& \\ 0& 1& 0& 1& 1& 0& 0& 1& \\ 0& 1& 1& 0& 1& 0& 1& 0& \\ 0& 0& 1& 0& 1& 1& 1& 0& \\ \end{pmatrix}$

How do I know if the rows of matrix are linearly independent? How to reduce the matrix so that all rows are linearly independent?

Answer 1

The submatrix formed by the first five columns of your matrix has determinant equal to $-2\neq0$. Therefore, the rows of your matrix are linearly independent.

Answer 2

Recall that the following statement are equivalent

• rows of matrix are linearly independent
• $rank(G')=5$
• we have no zero rows in the RREF