Determinant Identity for A-B and B A Matrices

[quasar987]

Hello matrices masters,

If A and B are nxn square matrices, is there an identity for the determinant of the block matrix

A -B
B A

?

Lots of thanks and praises.

 

[CompuChip]

Would http://www.mth.kcl.ac.uk/~jrs/gazette/blocks.pdf be able to help you out?

 

[Ben Niehoff]

The answer is also on Wikipedia: http://en.wikipedia.org/wiki/Determinant#Block_matrices

 

[quasar987]

Unfortunately, in the particular case I'm interested in, I don't see a reason to believe that A and B commute, nor that A or B is invertible. So none of the formulae on wiki are applicable.

I know for a fact that the determinant of that matrix is nonzero, and it is probably positive, which I am trying to show.

 

[daveyp225]

Did you try finding a pattern between the 2x2, 4x4, 9x9?