$A ^3= A$

$A ^2=\left[\begin{array}{lll}0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0\end{array}\right]\left[\begin{array}{lll}0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0\end{array}\right]$

$A ^2=\left[\begin{array}{ccc} a +2 & 2 c & 3 \\ 3 & a +3 c & 2 a \\ ac & 1 & 2+3 c \end{array}\right]$

$A ^3=\left[\begin{array}{ccc} a +2 & 2 c & 3 \\ 3 & a+3 c & 2 a \\ ac & a & 2+3 c \end{array}\right]\left[\begin{array}{ccc}0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0\end{array}\right]$

$A^3=\left[\begin{array}{ccc}2 a c+3 & a+2+3 c & 2 a+4+6 c \\ a(a+3 c)+2 a & 3+2 a c & 6+3 a+9 c \\ a+2+3 c & a c+c(2+3 c) & 2 a c+3\end{array}\right]$

Given $A ^3=A$

$2 ac +3=0 \ldots(1) \text { and } a+2+3 c=1$

$a +1+3 c =0$

$a +1-\frac{9}{2 a}=0$

$2 a ^2+2 a -9=0$

$f (1) < 0, f (2) > 0$

$a \in(1,2]$

$n=2$