If $A$ is a symmetric matrix, then $A^2$ is also symmetric
Andrew Henderson 
I first tried if the claim was true by testing it with a symmetric matrix, and I got that if I have a symmetric square matrix $A$ then $A^2$ is also symmetric. So to prove this for a general case I did:
First of all I take a general square matrix
$$A= \left[ {\begin{array}{ccc} a_{11} & a_{12} & ...a_{1n}\\ a_{21} & a_{12} & ...a_{1n}\\ ... & ... & ...\\ a_{n1} & a_{n2} & ...a_{nn}\\ \end{array} } \right]$$we can see that the matrix above is symmetric because it is equal to its transpose.
Then I calculate$$A^2= \left[ {\begin{array}{ccc} \sum\limits_{k=1}^n a_{1k}a_{k1} & \sum\limits_{k=1}^n a_{1k}a_{k2} & ...\sum\limits_{k=1}^n a_{1k}a_{kn}\\ \sum\limits_{k=1}^n a_{2k}a_{k1} & \sum\limits_{k=1}^n a_{2k}a_{k2} & ...\sum\limits_{k=1}^n a_{2k}a_{kn}\\ ... & ... & ...\\ \sum\limits_{k=1}^n a_{nk}a_{k1} & \sum\limits_{k=1}^n a_{nk}a_{k2} & ...\sum\limits_{k=1}^n a_{nk}a_{kn}\\ \end{array} } \right]$$so I get that $A^2$ is symmetric because it is equal to its transpose $(A^2)^T$ or we can say that because $a_{ij}a_{ji}=a_{ji}a_{ij}$ for all $1\le i,j\le n$.
Do you think this is a good proof or how can I improve it? Thanks
$\endgroup$ 12 Answers
$\begingroup$Why not directly?: We're given $\;A\;$ is symmetric $\;\iff A^t=A\;$ , and then
$$(A^2)^t=(AA)^t=A^tA^t=AA=A^2\implies A^2\;\;\text{is symmetric}$$
Or in short: $\;(A^2)^t=(A^t)^2=A^2\;$
$\endgroup$ 4 $\begingroup$More generally, let's prove to $A^n$ instead of $A^2$
Let's prove using induction on $n$ that $A=A^t \implies A^n = (A^n)^t$
Base case is $n=1$, that is $A = A^t$, that is true by hypothesis.
Assuming that is true for $n = k$ with $k \in \mathbb{N}$, we have $A^k = (A^k)^t$ , this is our inductive hypothesis.
We need prove it for $n=k+1$, that is, $A^{k+1}=(A^{k+1})^t$
And this is trivial to prove, since $A^{k+1}=A^k \cdot A=(A^k)^t \cdot A^t = (A^t)^k\cdot (A^t)= (A^t)^{k+1}=(A^{k+1})^t$
$$\Box$$
So our conclusion is that if $A$ is a symmetric matrix therefore $A^n$ is also symmetric.
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