Prove that if $A$ is a symmetric matric then $A^3$ and $A^2-2A+I$ are symmetric matrices.

Question

I am uncertain on how to approach this proof. For most everything I've encountered concerning symmetry, it has involved taking the transpose in order to show some property. Here, I'm not certain if and how that would be effective.

Answer 1

Hint: Use the fact that $(AB)^T = B^TA^T$.

Answer 2

Hint

To show that a matrix $B$ is symmetric, you have to show that $B^T=B$. Actually, you can easily show that if $A$ is symmetric, then $p(A)$ is also symmetric for any polynomial $P(X)\in\mathbb R[X]$.

Answer 3

Hint: $$(AB)^T=B^TA^T$$ and that the sum of symmetric matrices is also symmetric.