Determinants

[bloodhound]

how do u express a determinant of a sum or two matrices in terms of determinant of each matrix?

[MandrakeRoot]

As far as i remember there is no such formula.

 

Try for instance to see the link between the det(A + B), det(A) and det(B) when both A and B are n x n diagonal matrices ?

Now add two elements to B on the (1,2) and (2,1) position and see what happens with your formulae...

 

Mandrake

[matt grime]

There is no simple elementary formula, of the type that

 

det(A+B)= f(det(A),det(B))

 

for some simple f.

 

There is obviously a formula in terms of detA, detB and (some of) the entries in A and B, in fact there are obviously many such formulas. I don't know that any of them is easier (or less computationally intensive) to calculate than just working out det(A+B) the hard way purely in terms of the entries in A and B without using det(A) or det(B).

[matt grime]

Here is an example of how sums are usefully used in determinants.

 

Let A be nxn, and let A(i) be the matrix obtained by setting all entries in the first row of A to be zero except for the i'th, for 1<=i<=n, then

 

det(A)= det( sumA(i)) = sum det(A(i))

 

that is after all how we practically calculate the determinant by expanding about the first row.