If A and B are square matrices of the same order

Given:

$A$ and $B$ are square matrices of the same order

$A B=3 I$

Pre-Multiplying both sides by $A^{-1}$, we get

$\Rightarrow A^{-1}(A B)=A^{-1}(3 I)$

$\Rightarrow\left(A^{-1} A\right) B=3\left(A^{-1} /\right)$

$\Rightarrow(I) B=3 A^{-1}$

$\Rightarrow B=3 A^{-1}$

$\Rightarrow \frac{1}{3} B=A^{-1}$

Hence, if $A$ and $B$ are square matrices of the same order and $A B=3 I$, then $A^{-1}=\frac{1}{3} B$.