<s>[INST] $A^3 = I$ ($A$ is real Symmetric matrix). Does it imply that $A = I$? Question of our assignment
$A$ is a $3×3$ real symmetric matrix such that $A^3 = I$  (Identity matrix). Does it imply that $A = I$?
If so, why? If not, give an example.
Any help will be appreciated.
 [/INST] For any vector $x$, let $y=(A-I)x$. Since $A$ is symmetric and $A^3=I$, we have
\begin{align*}
\|Ay\|^2 + \|A^2y\|^2 + \|y\|^2
&=y^T\left[A^TA+(A^2)^TA^2+I\right]y\\
&=y^T\left(A^2+A^4+I\right)y\\
&=y^T(A^2+A+I)y\\
&=y^T(A^2+A+I)(A-I)x\\
&=y^T(A^3-I)x\\
&=0.
\end{align*}
Therefore $\|y\|$ must be zero, i.e. $y=0$ or $Ax=x$. Since $x$ is arbitrary, we conclude that $A=I$.

Edit. In general, suppose $A$ is real symmetric and $A^m=I$ for some odd positive integer $m$. For any vector $x$, let $y=(A-I)x$. Since $A$ is symmetric, we have $\|A^ry\|^2=(A^ry)^T(A^ry)=y^TA^{2r}y$ for every nonnegative integer $r$. It follows that
\begin{align*}
\|y\|^2
&\le\sum_{\substack{0\le k\le m-1\\ k\text{ is even}}}\|A^{k/2}y\|^2+\sum_{\substack{0\le k\le m-1\\ k\text{ is odd}}}\|A^{(k+m)/2}y\|^2\\
&=y^T\left(\sum_{\substack{0\le k\le m-1\\ k\text{ is even}}}A^k+\sum_{\substack{0\le k\le m-1\\ k\text{ is odd}}}A^{k+m}\right)y\\
&=y^T\left(\sum_{k=0}^{m-1}A^k\right)y\quad\text{(because $A^m=I$)}\\
&=y^T\left(\sum_{k=0}^{m-1}A^k\right)(A-I)x\\
&=y^T(A^m-I)x\\
&=0.\\
\end{align*}
Therefore $\|y\|$ must be zero, i.e., $y=0$ or $Ax=x$. Since $x$ is arbitrary, we conclude that $A=I$.
Similarly, if $A$ is real symmetric and $A^m=I$ for some even positive integer $m$, we may show that $A^2=I$. Indeed, if we put $y=(A^2-I)x$, then
\begin{align*}
\|y\|^2
&\le\|A^{(m-2)/2}y\|^2+\|A^{(m-4)/2}y\|^2+\cdots+\|Ay\|^2+\|y\|^2\\
&=y^T\left(A^{m-2}+A^{m-4}+\cdots+A^2+I\right)y\\
&=y^T\left(A^{m-2}+A^{m-4}+\cdots+A^2+I\right)(A^2-I)x\\
&=y^T(A^m-I)x\\
&=0\\
\end{align*}
and hence $y=(A^2-I)x$ must be zero for any vector $x$.
</s>