Determinant Theorem

Given a Matrix ${\hbox{\sf m}}$, the following are equivalent:

1. $\vert{\hbox{\sf m}}\vert \not=0$.

2. The columns of m are linearly independent.

3. The rows of m are linearly independent.

4. Range(m) = $\Bbb{R}^n$.

5. Null(m) = $\{0\}$.

6. m has a Matrix Inverse.

See also Determinant, Matrix Inverse, Nullspace, Range (Image)