determinant and invertibility

$\det A \ne 0$ exactly when a square matrix $A$ is invertible.

For a square matrix $A$ , $\det A \ne 0$ exactly when $A$ is invertible — the determinant produced by elimination dies exactly when elimination stalls on a zero pivot, which is exactly when $A$ fails to be invertible. This test only applies to square matrices; for a rectangular matrix, rank is the right tool instead.