Question

Prove that every invertible matrix possesses a unique inverse

Answer 1

Let A be the invertible matrix of order $\times m$ Let B and C be two inverses of A.

$AB=BA=Im$

and $AC=CA=I_m$

Now $AB=I_m\Rightarrow C\left(AB\right)=C{I}_n\Rightarrow \left(CB\right)B=C{I}_m$[By associativity]

$\Rightarrow I_{m}B=C{I}_m\Rightarrow B=C$