Why if $det(a) \ne 0$ then rows are linearly independent?

Question

Why if $det(a) \ne 0$ then rows are linearly independent? Trying find it in internet, but only found facts.

Answer 1

One way to look at it:

For a set of vectors forming a square matrix, to have $\det\neq 0$ means that this matrix is invertible. If the square matrix formed by your set of vectors is invertible, then you can transform this matrix into the identity matrix performing elementary row operations on it. This means that the vectors in the original matrix are linearly independent.