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I need extract $X$ But fail.

I tried several methods but still could not, I would be happy help.

$$A^t(X^{-1}+A)=X^{-1}$$ Thanks.

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    Do you know whether $A$ is invertible? – 2017-02-01
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    @Mathematician42 If $A$ is singular then the LHS cannot be equal to $X^{-1}$. – 2017-02-01
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    @ErickWong: Well sure, but I wanted the OP to think about that. – 2017-02-01

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You have the formula of $A^T(X^{-1} + A) = X^{-1}$ By multiplying the left hand side, and using right-multiplication with $X$ you get to $A^T(I+AX)=I$.

From there on, there is some more elementary matrix-manupilation to get to something like $X= \dots$.

Be aware, that the resulting $X$ does contain inverses of $A$.