Equivalence of Definitions of Singular Matrix

Theorem

The following definitions of the concept of Singular Matrix are equivalent:

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 $\struct {\map {\MM_R} n, +, \times}$.

Definition 1

Let $\mathbf A$ have no inverse.

Then $\mathbf A$ is referred to as singular.

Definition 2

Let the determinant of $\mathbf A$ be equal to $0$.

Then $\mathbf A$ is referred to as singular.

Proof

Follows directly from Matrix is Nonsingular iff Determinant has Multiplicative Inverse.

$\blacksquare$