Equivalence of Definitions of Non-Invertible Matrix

Theorem
Let $\struct {R, +, \circ}$ be a ring with unity.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $\mathbf A$ be an element of the ring of square matrices $\struct {\map {\MM_R} n, +, \times}$.

The following definitions for $\mathbf A$ to be non-invertible are equivalent:

Proof
Follows directly from Matrix is Invertible iff Determinant has Multiplicative Inverse.