Unquie solution i.e exactly one solution which is $Y \times A$

Given condition $A^2=I \Rightarrow A \times A=I \Rightarrow A=A^{-1}$

Now Y is $n$ dimensional vector .

$\therefore AX=Y \\ \Rightarrow X_{n \times 1}= A^{-1}_{n \times n} \times Y_{n \times 1} \\ \Rightarrow X=AY(solution) $

Given condition $A^2=I \Rightarrow A \times A=I \Rightarrow A=A^{-1}$

Now Y is $n$ dimensional vector .

$\therefore AX=Y \\ \Rightarrow X_{n \times 1}= A^{-1}_{n \times n} \times Y_{n \times 1} \\ \Rightarrow X=AY(solution) $