# Computing the determinant of a block matrix

4 views (last 30 days)
RickyBoy on 1 Aug 2022
Edited: RickyBoy on 1 Aug 2022
I am having trouble using a well-known formula for computing the determinant of a block matrix. That is, if
in which A and D are square matrices and exists then
.
The code snippet below should achieve this but it returns two different results. I'm sure it's a very trivial mistake I've made but I've probably been staring at it too long now to find it.
As a follow up, does anyone know if there is a suitably adjusted version of this formula for the case when it is B and C which are the square matrices?
function [detM,detBlock] = BlockDeterminant()
M = magic(4);
detM = det(M);
A = M(1:3,1:3);
B = M(1:3,4);
C = M(4,1:3);
D = M(4,4);
detBlock = det(A)*det(D-C*inv(A)*B);
end

KSSV on 1 Aug 2022
A = rand(2) ;
B = rand(2) ;
C = rand(2) ;
D = rand(2) ;
M = [A B; C D] ;
det(M)
ans = -0.0221
det(A)*det(D-C*inv(A)*B)
ans = -0.0221
In your case, D is a scalar.
Bruno Luong on 1 Aug 2022
Good. I don't get why contribution is so helpful, but if you say so...

Walter Roberson on 1 Aug 2022
Just round-off error. You are calculating two different order of operations and that affects the results .
If you use M = sym(magic(4)) you will get exact zero for each

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