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Math Help - Matrix determinant

  1. #1
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    Matrix determinant

    How is the best way to find a determinant for matrices of rows and columns more than 3?
    The question i need to solve is this:
    A=
    1211
    2210
    361u
    00uv

    Find the determinant of the matrix. Sketch in a u-v plane the set of points for which A has no inverse.

    What is a u-v plane? Please somebody help.
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  2. #2
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    You can use cofactors to find the determinant, but I would run it through a calculator that does them. My opinion about doing determinants by hand is kind of like Plato's about partial fractions and integrals. A calculator will do the grunt work in a second.

    The deteminant is 2(u^{2}-3u+2v)

    The matrix does not have an inverse if the determinant is 0.
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  3. #3
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    Thanks for your help, but i need to know how to do it for an exam. How did you figure it out?
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  4. #4
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    I ran it thorugh the calculator, but here's one way to do it the ol-fashioned way.

    It's not too bad. Interchange rows 3 and 4, then perform 2 elementary row operations to turn it into an upper trianguar matrix.

    Then multiply the diagonal.

    Interchange row 3 and 4:

    \begin{bmatrix}1&2&1&1\\2&2&1&0\\0&0&u&v\\3&6&1&u\  end{bmatrix}

    Multiply row 1 by -3 and add to row 4:

    \begin{bmatrix}1&2&1&1\\2&2&1&0\\0&0&u&v\\0&0&-2&u-3\end{bmatrix}

    2/u times row 3 added to row 4:

    \begin{bmatrix}1&2&1&1\\2&2&1&0\\0&0&u&v\\0&0&0&u+  \frac{2v}{u}-3\end{bmatrix}

    Now, multiply the diagonal:

    (1)(2)(u)(u+\frac{2v}{u}-3)=2(u^{2}-3u+2v)
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  5. #5
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    Do you happen to know how to sketch the u-v plane? I don't know what that is :S
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  6. #6
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    Quote Originally Posted by mathsstudent8 View Post
    Do you happen to know how to sketch the u-v plane? I don't know what that is :S
    What if you were asked to sketch x^{2}-3x+2y = 0 in the x-y plane ...... Capisce?
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  7. #7
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    Man i'm so going to fail this exam.

    Thanks for your help.
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  8. #8
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    Quote Originally Posted by mr fantastic View Post
    What if you were asked to sketch x^{2}-3x+2y = 0 in the x-y plane ...... Capisce?
    x^{2}-3x+2y = 0 \Rightarrow y = -\frac{x^2}{2} + \frac{3x}{2} is a parabola.

    Now just put back x --> u and y --> v to get that in the u-v plane the set of points you want lie on the parabola v = -\frac{u^2}{2} + \frac{3u}{2}.
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