Find Det (C) knowing Det(A) and Det(B)

  • Thread starter mmmboh
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  • #1
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Hi, right now I have two 3x3 matrices with letter entries, I know the determinant of A and the determinant of B, and I am given a matrix C for which I have to find the determinant for. Right now I am expressing matrix C as a combination of A and B, but what happens to the determinant when I add matrices A and B together?

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  • #2
Dick
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One of the rows of C is a linear combination of two vectors. det is linear in the row entries, right?
 
  • #3
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Well all the rows in C are a linear combination of vectors in A and B.
 
  • #4
Dick
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Well all the rows in C are a linear combination of vectors in A and B.

I'm referring to the second row being a sum of two vectors.
 
  • #5
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Ok, that's true, but how does that help me in finding the new determinant? I still have to know what happens when I add the matrices together, unless you mean something else?
 
  • #6
Dick
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I mean that if the letters A, B, C, and D are row vectors then det(A,B+D,C)=det(A,B,C)+det(A,D,C). det is linear in the row vectors.
 
  • #7
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Ah I didn't know you can do that :S...thanks!
 
  • #8
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But wait I don't understand how that works...why do the B and D add but nothing else...isn't this sort of like saying det(A+B)=det(A)+det(B) which isn't true?
 
  • #9
Dick
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But wait I don't understand how that works...why do the B and D add but nothing else...isn't this sort of like saying det(A+B)=det(A)+det(B) which isn't true?

No, it's not the same thing. I'm using that det is a multilinear function of it's rows (or columns). It's a fundamental property of the determinant. Try and find it in your book. It's important.
 

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