# Linear Algebra: Determinant

1. Feb 21, 2012

### drosales

I'm having trouble with this problem on my homework

Let n be a positive integer and A=[ai,j] A is n*n. Let B=[Bi,j] B is n*n be the matrix defined by bi,j=(-1)i+j+1 for 1<i,j<n. Show that det(B)=(-1)ndet(A)

Hint: use the definition of determinant

I honestly have no idea how to go about this. I'm assuming it has something to do with elementary row operations and the sign of the determinant changing with each operation but am not quite sure how to get started. Any help would be great.

2. Feb 21, 2012

### Fredrik

Staff Emeritus
A good start would be to write down the definition of the determinant. Specifically, what does the definition say that $\det B$ is?

Did you mean $b_{ij}=(-1)^{i+j+1}a_{ij}$?

3. Feb 21, 2012

### drosales

Yes, that is what was meant. I didnt realize I didnt complete that

4. Feb 21, 2012

### Fredrik

Staff Emeritus
So what does "det B" mean for an arbitrary n×n matrix?

5. Feb 21, 2012

### drosales

My understanding is that det(B) is the sum of the cofactor expansions multiplied by minor matrices

6. Feb 21, 2012

### Fredrik

Staff Emeritus
This is usually derived from a definition involving a sum over all permutations of {1,...,n}. I think that definition will be easier to work with.

7. Feb 21, 2012

### drosales

Would you mind explaining it? I have it in my lecture notes but I have trouble following

8. Feb 21, 2012

### Fredrik

Staff Emeritus