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.

$\blacksquare$