NOTE: I was not able to upload the matrix in the normal way so I created the image and i think it is easy to view it now .
2026-04-01 21:31:16.1775079076
intuition behind finding inverse of this matrix
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You can write $$\begin{bmatrix}A&0\\B&C\end{bmatrix}\begin{bmatrix}X&Y\\Z&W\end{bmatrix}=\begin{bmatrix}I&0\\0&I\end{bmatrix}$$ Now you have 4 equations to solve: $$\begin{align} AX &= I\\ AY &= 0\\ BX+CZ &= 0\\ BY+CW &= I \end{align}$$ So, from (1) and (2) you get $X=A^{-1}$ and $Y=0$ and other results follow similarly.