linear algebra - find x

Question

Can you help solve for $X$?

I tried several times but could not. I'd love to get some help.

$$A^T\left(X^{-1}+A\right)=X^{-1}$$

where

$$A=\begin{pmatrix}1&1&-1\\0&-1&1\\-1&0&1\end{pmatrix}.$$

Answer 1

If the matrices are invertible

$$A^TX^{-1}+A^TA=X^{-1},$$

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

$$I=(A^TA)^{-1}(I-A^T)X^{-1},$$ $$X=(A^TA)^{-1}(I-A^T)=A^{-1}A^{-T}(I-A^T)=A^{-1}A^{-T}-A^{-1}=A^{-1}(A^{-T}-I).$$

Answer 2

$A^T(X^{-1}+A)=X^{-1}$ multiply by $(A^T)^{-1}$ on the left:

$X^{-1}+A=(A^T)^{-1}X^{-1}$; then substract $X^{-1}$:

$A=(A^T)^{-1}X^{-1}-X^{-1}= ((A^T)^{-1}-I)X^{-1}$; multiply by $X$ on right:

$AX = (A^T)^{-1}-I$ multiply by $A^{-1}$ on the left:

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

Calculating you get $X= \begin{pmatrix} 1&0&2 \\ 0&2&1 \\ 1&1&2 \end{pmatrix}$.