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How are the determinants of A and B related? (Do not compute det(A))

  1. Dec 3, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider the matrices

    A =
    a1 a2 a3
    b1 b2 b3
    c1 c2 c3
    and

    B =
    3a1 4a2+5a1 6a3
    3b1 4b2+5b1 6b3
    3c1 4c2+5c1 6c3

    How are the determinants of A and B related? DO NOT COMPUTE det(A)!

    2. Relevant equations



    3. The attempt at a solution

    I'm completely lost... I tried doing Kramer rule for det(B) but don't understand how I'm supposed to do this. I mean, I can see that the 2nd column of B is equal to 5 times the first plus 4 times the second columns of A, that the 1st column of B is 3 times the first column of A and that the 3rd column of B is 6 times the third column of A. But what am I meant to draw from this?
     
  2. jcsd
  3. Dec 3, 2012 #2

    jedishrfu

    Staff: Mentor

    I defer to D H
     
    Last edited: Dec 3, 2012
  4. Dec 3, 2012 #3

    D H

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    Staff Emeritus
    Science Advisor

    Almost there. Try to express this in matrix form. In other words, see if you can find a matrix C such that either C*A=B or A*C=B.

    What's the determinant of a product of two matrices?
     
  5. Dec 3, 2012 #4

    Ray Vickson

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    Science Advisor
    Homework Helper

    You are supposed to use standard results about determinants obtained by adding multiples of row (or columns) to other rows (or columns). I will let you find these, because the exercise of searching will help you to remember them.
     
  6. Dec 3, 2012 #5
    oh, is it that simple?

    3*5*6det(A)=det(B)?
     
  7. Dec 3, 2012 #6

    Ray Vickson

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    Almost. Where does the factor '5' come from?
     
  8. Dec 3, 2012 #7
    sorry. typo. i meant to write a 4. adding columns/rows into others dont change the determinant, but multiplying a row/column multiplies the determinant by the same factor.

    3*4*6

    72det(A)=det(B)
     
  9. Dec 3, 2012 #8

    Ray Vickson

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    Science Advisor
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    Right!
     
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