The entries of diagonal matrix $D$ are only $1$ or $-1$ then $D=D^{-1}$

Question

The entries of diagonal matrix $D$ are only $1$ or $-1$ then $D=D^{-1}$

My attempt: First I thought maybe determinant works but it does not because it is not certain that determinant is equal to $1$ or $-1$ and also that gives us nothing about entries for $D^{-1}$.

Intuitively, I can say yes that should be true but I couldn't prove. I ask myself the following how do I find inverse of matrix? One way is using r.r.e.f. Since matrix has only $+1$ and $-1$ then its r.r.e.f should be itself.

Is there any other way to show?

Answer 1

Claim: $D=\textrm{diag}(d_1, d_2, \ldots ,d_n) $ and $d_i\neq 0 ,\forall i\in\Bbb{N}_n$ then $D^{-1}=\textrm{diag}(\frac{1}{d_1}, \frac{1}{d_2}, \ldots ,\frac{1}{d_n})$

Hint: Compute $DD^{-1}$