Equivalence of Definitions of Non-Invertible Matrix
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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:
Definition 1
Let $\mathbf A$ have no inverse.
Then $\mathbf A$ is referred to as non-invertible.
Definition 2
Let the determinant of $\mathbf A$ be equal to $0$.
Then $\mathbf A$ is referred to as non-invertible.
Proof
Follows directly from Matrix is Invertible iff Determinant has Multiplicative Inverse.
$\blacksquare$