Transformation of inverse to a system of linear equations. Need to solve $X=({U}^{\prime}WU{)}^{-1}{U}^{\prime}$. ${U}^{\prime}$ is ${U}^{\prime}$ is $7\times 7$ positive definite matrix, ${U}^{\prime}$ is of rank $$3$$.

Transformed $({U}^{\prime}WU{)}^{-1}{U}^{\prime}$ as

$({U}^{\prime}WU{)}^{-1}{U}^{\prime}WU=I\phantom{\rule{0ex}{0ex}}XWU=I\phantom{\rule{0ex}{0ex}}{U}^{\prime}W{X}^{\prime}=I\phantom{\rule{0ex}{0ex}}(I\otimes {U}^{\prime}W)vec({X}^{\prime})=vec(I).\phantom{\rule{0ex}{0ex}}$

When I solved $X=({U}^{\prime}WU{)}^{-1}{U}^{\prime}$ and as the above linear system using $$R$$, the answers are slightly different. Does something wrong with the above logic?

Transformed $({U}^{\prime}WU{)}^{-1}{U}^{\prime}$ as

$({U}^{\prime}WU{)}^{-1}{U}^{\prime}WU=I\phantom{\rule{0ex}{0ex}}XWU=I\phantom{\rule{0ex}{0ex}}{U}^{\prime}W{X}^{\prime}=I\phantom{\rule{0ex}{0ex}}(I\otimes {U}^{\prime}W)vec({X}^{\prime})=vec(I).\phantom{\rule{0ex}{0ex}}$

When I solved $X=({U}^{\prime}WU{)}^{-1}{U}^{\prime}$ and as the above linear system using $$R$$, the answers are slightly different. Does something wrong with the above logic?