Equivalence of Definitions of Non-Invertible Matrix

Theorem
Let $\left({R, +, \circ}\right)$ be a ring with unity.

Let $\mathcal M_R \left({n}\right)$ be the $n \times n$ matrix space over $R$.

Let $\mathbf A$ be an element of the ring $\left({\mathcal M_R \left({n}\right), +, \times}\right)$.

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

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