Assume that an orthogonal projection matrix is given as $$ A = \left[ X(X'\Sigma^{-1}X)^-X'\Sigma^{-1}\right]$$ where $\Sigma$ is positive definite and $X_{m\times n}$ is design matrix. Show that $A$ is the same for any choice of $(X' \Sigma^{-1} X)^-.$
My Idea: Let $G = (X' \Sigma X)^-$ be a generalized inverse of $A$ and since $G$ is not unique, we assume $G_\star$ to be another generalized inverse of $A$ such that $$AG_\star A = A$$ so that we could have \begin{align} [A]G_\star [A] & = \left[ X(X'\Sigma^{-1}X)^-X' \Sigma^{-1} \right] (X' \Sigma^{-1} X)_\star^- \left[ X(X'\Sigma^{-1}X)^-X\Sigma^{-1} \right] \\ & = \left[ X(X'\Sigma^{-1}X)^-X' \Sigma^{-1} \right] X^{'-}_\star \Sigma _\star X_\star^- \cdot \left[ X X'^- \Sigma X ^-X\Sigma^{-1} \right] \\ & = \left[ X(X'\Sigma^{-1}X)^-X' \Sigma^{-1} \right] \left[(X_\star'^-X')\cdot (\Sigma_\star\Sigma^{-1}) \cdot(X_\star^-X) \cdot (X^- X) \cdot \Sigma \right]\\ & = \left[ X(X'\Sigma^{-1}X)^-X' \Sigma^{-1} \right] \left[I \cdot I \cdot I \cdot I \cdot \Sigma \right]\\ & = \left[ X(X'\Sigma^{-1}X)^-X' \Sigma^{-1} \right] \Sigma\\ & \neq A \end{align}
I probably did something incorrect. What am I missing?