Properties of matrix inverse

Question

Suppose $A$ is a $l \times l$ matrix and $\Gamma$ is a $l \times k$ matrix. Is it true that $\Gamma[\Gamma'A\Gamma]^{-1}\Gamma' = A^{-1}$, assuming $A^{-1}$ exists? If yes, how to show it?

Answer 1

There are clearly counterexamples: for example take $\Gamma$ to be the zero matrix. Then $\Gamma'A\Gamma$ is zero, so is not invertible, so the left hand side of your equation doesn't exist.

If we add an assumption that $\Gamma'A\Gamma$ is invertible, then, in particular, we must have $k \geq l$ (the dimension of the image is at most $k$), and $k \leq l$ (the dimension of the kernel is at least $k - l$), so $l = k$, and $\Gamma$ is an invertible $l\times l$ matrix, so we have $\Gamma[\Gamma'A\Gamma]^{-1}\Gamma' = \Gamma[\Gamma^{-1}A^{-1}\Gamma'^{-1}]\Gamma' = (\Gamma\Gamma^{-1})A^{-1}(\Gamma'^{-1}\Gamma') = A^{-1}$.