Is the inverse of a symmetric matrix also symmetric?

Let $A$ be a symmetric invertible matrix, $A^T=A$, $A^{-1}A = A A^{-1} = I$ Can it be shown that $A^{-1}$ is also symmetric?

I seem to remember a proof similar to this from my linear algebra class, but it has been a long time, and I can’t find it in my text book.

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In fact, $(A^T)^{-1}=(A^{-1})^T$. Indeed, $A^T(A^{-1})^T=(A^{-1}A)^T=I$.

You can’t use the thing you want to prove in the proof itself, so the above answers are missing some steps. Here is a more complete proof. Given A is nonsingular and symmetric, show that $ A^{-1} = (A^{-1})^T $:

$$ I = I^T $$

since $ AA^{-1} = I $,

$$ AA^{-1} = (AA^{-1})^T $$

since $ (AB)^T = B^TA^T $,

$$ AA^{-1} = (A^{-1})^TA^T $$

since $ AA^{-1} = A^{-1}A = I $, we rearrange the left side

$$ A^{-1}A = (A^{-1})^TA^T $$

since $ A = A^T $, we substitute the right side

$$ A^{-1}A = (A^{-1})^TA $$
$$ A^{-1}A(A^{-1}) = (A^{-1})^TA(A^{-1})$$
$$ A^{-1}I = (A^{-1})^TI $$
$$ A^{-1} = (A^{-1})^T $$

and we are done.

Yes.

$$
AB=BA=I\quad\Rightarrow\quad B^TA^T=A^TB^T=I\quad\Rightarrow\quad B^TA=AB^T=I
$$

Another way to see that is to recall the formula $$A^{-1} = \frac{1}{\det(A)} \mathrm{Adj}(A)^T$$ and to note that the adjoint matrix of a symmetric matrix is by construction symmetric.

Yes. The inverse $A^{-1}$ of invertible symmetric matrix is also symmetric:

\begin{align}
A & = A^T \tag{Assumption: $A$ is symmetric}\\ \\
A^{-1} & = (A^T)^{-1} \tag{$A$ invertible $\implies A^T = A$ invertible}\\ \\
A^{-1} & = (A^{-1})^T \tag{identity: $(A^T)^{-1} = (A^{-1})^T$} \\ \\
& {\large \therefore}\quad \rlap{\text{If $A$ is symmetric and invertible, then $A^{-1}$ is symmetric}}
\end{align}

We have two properties to use: $A^{T}=A$ and $A^{-1} $exist, here we go!

$$
A^{-1}A=I
$$since $A^{-1}$exist
$$
(A^{-1}A)^{T}=A^{T}(A^{-1})^{T}=A(A^{-1})^{T}=I^{T}=I
$$since $A^{T}=A$
$$
A^{-1}A(A^{-1})^{T}=A^{-1}I
$$left multiple by $A^{-1}$

Thus
$$I(A^{-1})^{T}=(A^{-1})^{T}=A^{-1}$$

$$
(AA^{-1})^T=I^T=I=(A^{-1})^TA^T=(A^{-1})^TA=I\quad\Rightarrow\quad(A^{-1})^T=IA^{-1}\quad\Rightarrow\quad(A^{-1})^T=A^{-1}
$$
By definition
$$A=A^T$$
Used linear algebra equations
$$(AB)^T=B^TA^T;AA^{-1}=I;I^T=I$$