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Linear Algebra: Determinant

by drosales
Tags: algebra, determinant, linear
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drosales
#1
Feb21-12, 10:13 AM
P: 9
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.
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Fredrik
#2
Feb21-12, 10:19 AM
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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}##?
drosales
#3
Feb21-12, 10:26 AM
P: 9
Yes, that is what was meant. I didnt realize I didnt complete that

Fredrik
#4
Feb21-12, 10:28 AM
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Linear Algebra: Determinant

So what does "det B" mean for an arbitrary nn matrix?
drosales
#5
Feb21-12, 10:39 AM
P: 9
My understanding is that det(B) is the sum of the cofactor expansions multiplied by minor matrices
Fredrik
#6
Feb21-12, 10:41 AM
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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.
drosales
#7
Feb21-12, 11:06 AM
P: 9
Would you mind explaining it? I have it in my lecture notes but I have trouble following
Fredrik
#8
Feb21-12, 11:16 AM
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I think you have to be more specific about what's causing you trouble. I mean this definition:
http://en.wikipedia.org/wiki/Determi...-by-n_matrices


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