Definition of Inverse Matrix of a Square Matrix

Definition 1.2 (inverse of a square matrix). Let $A$ be a square matrix of order $n$. If there exists a square matrix $B$ of order $n$ with

then $B$ is called an inverse of $A$, written $A^{-1}$.

Theorem 1.2 (uniqueness). If a square matrix has an inverse, it is unique. Proof. Suppose $B$ and $C$ are both inverses of $A$: $AB=BA=I_n$ and $AC=CA=I_n$. Then $C=CI_n=C(AB)=(CA)B=I_nB=B$, so $B=C$.

Theorem 1.3 (existence -- the working formula). $A^{-1}$ exists if and only if $A$ is non-singular.

Proof (necessity). If $A^{-1}$ exists, $AA^{-1}=I_n$, so $|A||A^{-1}|=|I_n|=1$ by the determinant product rule -- forcing $|A|\ne0$.

Proof (sufficiency, and the formula). If $A$ is non-singular, $|A|\ne0$, and Theorem 1.1 gives $A(\operatorname{adj}A)=(\operatorname{adj}A)A=|A|I_n$. Dividing by the non-zero scalar $|A|$,

so $B=\frac1{|A|}\operatorname{adj}A$ satisfies the definition of an inverse. Hence

Remark. A singular matrix ($|A|=0$) has no inverse at all -- there is no matrix $B$ that can satisfy $AB=BA=I_n$, since that would force $|A|\ne0$ by the necessity argument above. …