Find a mistake game! Matrix determinant using block matrices

Question

I need your help finding where is my mistake while calculating 5x5 matrix determinant using block matrices.

My calculations are

I tripple checked everything and the online calculation of matrix A gives determinant of 4. And I get -8 using block matrices.

I just can't see why am I getting wrong answer.

Please help!

EDIT: The formula I am using is: det(AD-ACA^(-1)B) A being

0 1
1 0

B is

0 -2 1
3 1 1

C

1 -1
2 2
3 1

D

1 1 1
1 0 1
1 1 2

Det of part A is -1.

Answer 1

The error is in the first step. If you write your matrix as $\begin{pmatrix}A & B \\ C & D\end{pmatrix}$, you claim that the determinant equals $\textrm{det}(-D-CB)$. However, the correct formula is $\textrm{det}(A)\textrm{det}(D-CA^{-1}B)$.

Answer 2

You're using the formula $$\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\det(AD-BC)$$ where $A,B,C,D$ are not square matrices.

This is false in general. The formula holds if they're square matrices(in particular) and there is a commutation condition between $C$ and $D$.