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Homework Help: [Linear Algebra] Showing equality via determinant properties

  1. Sep 15, 2010 #1
    Problem:

    Show, without evaluating directly, that
    [tex]
    \left|\begin{matrix}
    a_1+b_1t&a_2+b_2t&a_3+b_3t \\
    a_1t+b_1&a_2t+b_2&a_3t+b_3 \\
    c_1&c_2&c_3 \end{matrix}\right|
    =
    (1-t^2)\left|\begin{matrix}
    a_1&a_2&a_3 \\
    b_1&b_2&b_3 \\
    c_1&c_2&c_3 \end{matrix}\right|
    [/tex]

    Clearly, here I'm supposed to use the determinant properties, do some row ops on the first array, and end up with the RHS.

    1. -tR1-R2 -> Row2 (no coefficient on determinant).

    [tex]
    \left|\begin{matrix}
    a_1+b_1t&a_2+b_2t&a_3+b_3t \\
    b_1-b_1t^2&b_2-b_2t^2&b_3-b_3t^2 \\
    c_1&c_2&c_3 \end{matrix}\right|
    [/tex]

    From here I can see that [tex]b_1-b_1t^2=b_1(1-t^2)[/tex]. But multiplying R2 by [tex]\frac{1}{1-t^2}[/tex] means I have to also pull that out as a coefficient to the entire array. So now:
    [tex]
    \frac{1}{1-t^2}
    \left|\begin{matrix}
    a_1+b_1t&a_2+b_2t&a_3+b_3t \\
    b_1&b_2&b_3 \\
    c_1&c_2&c_3 \end{matrix}\right|
    [/tex]

    Another row op, R1 -> R1-tR2, and I have my RHS, except the coefficient is reciprocated. Am I doing this wrong?
     
  2. jcsd
  3. Sep 15, 2010 #2

    ehild

    User Avatar
    Homework Helper

    If you multiply a row or column by some t, the value of the determinant becomes t times the original value. So you need to divide the whole determinant by t so as to keep it unchanged.

    ehild
     
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