Results 1 to 6 of 6

Math Help - Confusing Matrix Question

  1. #1
    Super Member craig's Avatar
    Joined
    Apr 2008
    Posts
    748
    Thanks
    1
    Awards
    1

    Confusing Matrix Question

    Let an n \times n determinant be:

    C_n =  \left| \begin{array}{cccc} \frac{1}{a_1 + b_1} & \frac{1}{a_1 + b_2} & ... & \frac{1}{a_1 + b_n} \\ \frac{1}{a_2 + b_1} & \frac{1}{a_2 + b_2} & ... & \frac{1}{a_2 + b_n} \\ ... & ... & ... & ...\\ \frac{1}{a_n + b_1} & \frac{1}{a_n + b_2} & ... & \frac{1}{a_n + b_n} \\ \end{array} \right|

    In the question we are also given that:

    C_n = \frac{\Pi_{j>k}(a_j-a_k)(b_j-b_k)}{\Pi_{j,k}(a_j+b_k)}

    Where \Pi_{j,k} symbolises a product over all possible j and k and \Pi_{j>k} a product subject to j > k.

    Prove the above equation for both n =2 and n=3.

    Have got no idea where to start with this, I'm not even sure if I understand the equation itself?

    If someone could help me with proving the n =2 case I'm sure I could figure out the n =3 part.

    Thanks in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Black's Avatar
    Joined
    Nov 2009
    Posts
    105
    For n = 2, we have

    C_2=\left|\begin{array}{cc}\frac{1}{a_1+b_1}&\frac  {1}{a_1+b_2} \\ \frac{1}{a_2+b_1}&\frac{1}{a_2+b_2}\end{array}\rig  ht|

    =\frac{1}{(a_1+b_1)(a_2+b_2)}-\frac{1}{(a_1+b_2)(a_2+b_1)}

    =\frac{(a_1+b_2)(a_2+b_1)-(a_1+b_1)(a_2+b_2)}{(a_1+b_1)(a_2+b_2)(a_1+b_2)(a_  2+b_1)}

    =\frac{a_2b_2-a_2b_1-a_1b_2+a_1b_1}{(a_1+b_1)(a_2+b_2)(a_1+b_2)(a_2+b_1  )}

    =\frac{(a_2-a_1)(b_2-b_1)}{(a_1+b_1)(a_2+b_2)(a_1+b_2)(a_2+b_1)}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member craig's Avatar
    Joined
    Apr 2008
    Posts
    748
    Thanks
    1
    Awards
    1
    Ahh I see thankyou! I'll attempt the 3x3 in the morning and post up my (hopefully correct) work

    Thanks again
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member craig's Avatar
    Joined
    Apr 2008
    Posts
    748
    Thanks
    1
    Awards
    1
    Finally got round to calculating the 3x3 determinant, and - with a little help from Maple - got:

    \frac{(a_1 - a_2)(a_1 - a_3)(a_2 - a_3)(b_1 - b_2)(b_1 - b_3)(b_2 - b_3)}{(a_1 + b_1)(a_2 + b_2)(a_3 + b_3)(a_2 + b_3)(a_3 + b_2)(a_1 + b_2)(a_2 + b_1)(a_3 + b_1)(a_1 + b_3)}

    This looks ok doesn't it?

    Thanks again for the help, wouldn't have been able to start without it.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member Black's Avatar
    Joined
    Nov 2009
    Posts
    105
    Yep, it equals to

    \frac{(a_2-a_1)(b_2-b_1)(a_3-a_1)(b_3-b_1)(a_3-a_2)(b_3-b_2)}{(a_1+b_1)(a_1+b_2)(a_1+b_3)(a_2+b_1)(a_2+b_2  )(a_2+b_3)(a_3+b_1)(a_3+b_2)(a_3+b_3)},

    which coincides with the formula for C_n.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member craig's Avatar
    Joined
    Apr 2008
    Posts
    748
    Thanks
    1
    Awards
    1
    Thanks again
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Confusing Question
    Posted in the Algebra Forum
    Replies: 4
    Last Post: January 30th 2009, 06:07 PM
  2. confusing calculus 1 question
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 27th 2009, 10:10 AM
  3. Matrix multiplication, confusing proof
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 20th 2008, 11:31 AM
  4. Confusing question
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: October 2nd 2008, 03:01 AM
  5. Confusing Question
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: October 4th 2006, 04:42 PM

Search Tags


/mathhelpforum @mathhelpforum