Using the Determinant to verify Linear Independence, Span and Basis

Most introductory books on Linear Algebra have a Theorem which says something like

Let $A$ be a square $n \times n$ matrix. Then the following are equivalent:

• $A$ is invertible.
• $\det(A) \neq 0$.
• The columns of $A$ are linearly independent.
• The columns of $A$ span $R^n$.
• The columns of $A$ are a basis in $R^n$.
• The rows of $A$ are linearly independent.
• The rows of $A$ span $R^n$.
• The rows of $A$ are a basis in $R^n$.

• The reduced row echelon form of $A$ has a leading 1 in each row.

  and many other conditions.....

What does this mean, it simply means that if you want to check if any of these conditions is true or false, you can simply pick whichever other condition from the list and check it instead..

Your question is: Can instead of third or fourth condition, check the second? That's exactly what the Theorem says: YES.