Question

If A is a symmetric matrix, then A³ is a ___________. matrix.

Answer 1

It is given that, A is symmetric matrix.

$\therefore A^T=A$ .....(1)

Now,

${\left(A^3\right)}^T$

$={\left(A^T\right)}^3$ $\left[{\left(A^n\right)}^T={\left(A^T\right)}^n,\forall n\in \mathrm{N}\right]$

$=A^3$ [Using (1)]

Since ${\left(A^3\right)}^T$$=A^3$, so the matrix A³ is symmetric.

If A is a symmetric matrix, then A³ is a __symmetric__ matrix.