Suppose $Y$ is a symmetric block matrix with 4 blocks
$$Y = \begin{bmatrix} aX & bX \\ bX & cX \end{bmatrix} $$ where $a, b, c, d$ are constants, and $X$ is an invertible submatrix.
Using the 2x2 block matrix inverse formula,
$$Z := Y^{-1} = \begin{bmatrix} \frac{ac-b^2}{c}X & \frac{-b(ac-b^2)}{c^2}X \\ \frac{-b(ac-b^2)}{c^2}X & \frac{ac-b^2}{a}X \end{bmatrix} $$
If I apply the 2x2 block matrix inverse formula again, I get
$$Z^{-1} = \begin{bmatrix} \frac{c^3(ac-b^2)-ab^2(ac-b^2)}{c^4}X & \frac{c^3(ac-b^2)-ab^2(ac-b^2)-abc^3}{c^4} X \\ \frac{c^3(ac-b^2)-ab^2(ac-b^2)-abc^3}{c^4}X & \frac{c^4(ac-b^2)-ab^2(ac-b^2)}{ac^4}X \end{bmatrix} $$
However, shouldn't $Y = Z^{-1}$? Where did I go wrong?