# Matrix determinant operator commutative?

1. Aug 4, 2009

### el_llavero

I've been going through properties of determinants of matrices and found the following:

Assuming products are defined and the matrices involved are nonsingular of the same order

The determinant of the product of any number of matrices is equal to the determinant of each matrix; where the order of factors does not matter

det(AB)=det(A)det(B)
det(AB)=det(B)det(A)
det(BA)=det(A)det(B)
det(BA)=det(B)det(A)

det(ABC)=det(C)det(A)det(B)

det(ACB)=det(A)det(B)det(C)

Is this correct? And is there a way to describe this property regarding commutatively? I know in general matrix multiplication is not commutative unless the matrices involved are diagonal and of the same dimension. However the determinant operator seems to not preserve the non commutative property of matrix multiplication, on either side of the equality. What I’m looking for here is a formal way of describing this property that encompasses the fact that order of factors does not matter and if commutativity should be used in any part of this description.

Last edited: Aug 4, 2009
2. Aug 4, 2009

### CompuChip

Probably (multiplicative) distributivity for the determinant together with commutativity of it's target space are the terms you are looking for.

3. Aug 4, 2009

### el_llavero

Thanks, that makes sense.

4. Aug 4, 2009

### el_llavero

Something isn't completely accurate in the second sentence of the original post

"Assuming products are defined and the matrices involved are nonsingular of the same order"

since it doesn't matter if the matrices are nonsingular for this property to hold a more accurate statement would be

"Assuming products are defined and the matrices involved are square matrices"