# Determinant of a matrix with identity blocks

1. Feb 15, 2010

### penguin007

Hi all,
I'm studying my mathematics lesson, and there is an example I can't understand:

Consider the matrix
A=(0 In)
(-In 0)
with In the identity nxn
We want to compute detA :

We introduce the permutation
p=(1 2 ... n n+1 ... 2n)
(n n+1 ... 2n 1 ... n )

s.t if we apply p to the columns of A, then we get
B=(In 0)
(0 -In)
(so far so good)

then we say: detB=(-1)^n (ok)
the sign of p is (-1)^n (all right)

then detA=(-1)^n*(-1)^n=1...(why???)

Last edited: Feb 16, 2010
2. Feb 15, 2010

### tiny-tim

Hi penguin007!

PA = B, so detP*detA = detB

3. Feb 15, 2010

### vela

Staff Emeritus
Re: matrix-determinant

How'd you get the sign of p is $(-1)^n$? That's where the problem lies.

4. Feb 16, 2010

### penguin007

Re: matrix-determinant

Hi tiny-tim:
I think I got it but I thought a map could be defined by a matrix only if it was linear?? Yet I can see p has a matrix (the one that reverses the columns 1 and n...) so:
is p a linear map? or can some non linear maps have a matrix??

vela:
s=The sign of p is given by (-1)^(the number of inversions by p) I counted n-2 inversions then s=(-1)^n...

Thanks again

5. Feb 16, 2010

### tiny-tim

Hi penguin007!
P is linear

what makes you think it isn't?

6. Feb 16, 2010

### vela

Staff Emeritus
Re: matrix-determinant

Oy, I misread what you wrote in this thread too. I think I'll just shut up now.

7. Feb 16, 2010

### penguin007

Re: matrix-determinant

tiny-tim:
please correct me: p is a permutation; p is linear would mean, for instance:p(3)=p(1+2)=p(1)+p(2), but p(3)=n+2<>p(1)+p(2)...( I know there is a snag, but where??)

vela:
That's all right, these things happen to everyone.

8. Feb 16, 2010

### tiny-tim

oh i see …

no, these Ps acts on n-tuples (or vectors).

So eg P12(a,b,c,d,e,f) = (b,a,c,d,e,f),

and you can check that P12(a+A,b,c,d,e,f) = (b,a,c,d,e,f) + (b,A,c,d,e,f) = P12(a,b,c,d,e,f) + P12(A,b,c,d,e,f)

9. Feb 16, 2010

### penguin007

Re: matrix-determinant

Thanks a lot tiny-tim for all these explanations, now I understand!