The four statements below are equivalent for a square matrix $A$, each implies the next in a natural cycle of reasoning. Order the chain starting from the determinant.

1. $\det A \ne 0$: the transformation preserves some volume
2. So no direction is collapsed: $Ax = 0$ only for $x = 0$
3. So the columns of $A$ are linearly independent
4. So every equation $Ax = b$ has exactly one solution

This chain is a slice of the Invertible Matrix Theorem, a dozen conditions that live or die together. One number, $\det A$, answers all of them at once, which is why "check the determinant" is the first move on any square system.