Question:
If $A$ is a non-singular square matrix such that $A^{3}=I$, then $A^{-1}=$___________
Solution:
Given: $A^{3}=1$
$A^{3}=1$
Multiplying both sides by $A^{-1}$, we get
$\Rightarrow A^{3} A^{-1}=I A^{-1}$
$\Rightarrow A^{2}\left(A A^{-1}\right)=I A^{-1}$
$\Rightarrow A^{2}(I)=A^{-1}$
$\Rightarrow A^{2}=A^{-1}$
Hence, if $A$ is a non-singular square matrix such that $A^{3}=l$, then $A^{-1}=\underline{A}^{2}$.