If A is a non-singular square matrix

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 $A^{-1}=\underline{A}^{2}$.

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