If A is an n×n invertible symmetric matrix, then (A^(-1) )^k is symmetric, for any positive integer k.

Question

i have this t/f question with prove required.

If $A$ is an $n×n$ invertible symmetric matrix, then $(A^-$$^1 )^k$ is symmetric, for any positive integer $k$.

Answer 1

Firstly, note that the inverse of a symmetric matrix is also symmetric. I.e. if $A$ is an invertible symmetric matrix, then $A^{-1} = {A^{-1}}^T$. (Is the inverse of a symmetric matrix also symmetric?)

Thus, the question can be simplified to:

If a matrix $A$ is invertible and symmetric, are all of its exponents $A^k$ also symmetric?

The answer to this question is yes as well. You can see this through induction:

For the base case, we have $AA = A^T A^T = (AA)^T$

Now, assume $A^k = (A^k)^T$.

Then $A^{k+1} = A^k A = (A^k)^T A^T = (A A^k)^T = (A^{k+1})^T$

Q.E.D

Thus, the statement

If A is an n×n invertible symmetric matrix, then (A−1)k is symmetric, for any positive integer k.

is True.