# Find a mistake game! Matrix determinant using block matrices

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.

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.

• Can you describe your approach? What formula are you using? – Exodd Jan 4 '17 at 16:19
• I have inserted your calculations as an inline image. – Andreas Caranti Jan 4 '17 at 16:23
• You can't replace $A$ with $-1$ to end up with $-D\pm CB$. – LinAlg Jan 4 '17 at 16:37
• Yes, I am trying to get this correctly this time. – Deramite Jan 4 '17 at 16:42

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 $\textrm{det}(A)\textrm{det}(D-CA^{-1}B)$.
• With $-D+CB$ you end up with the wrong answer though. – LinAlg Jan 4 '17 at 16:36
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 $D$.