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Math Help - Find unkown value for matrices

  1. #1
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    Find unkown value for matrices

    I am new at matrices and algebra. Kindly help me out to solve the problem from the first chapter of my textbook.

    For a lower triangular matrix L and an upper triangular matrix U of order three with unit diagonal elements determine P such that
    (a) LP = PL = I
    (b) UP = PU = I
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  2. #2
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    Re: Find unkown value for matrices

    To transform a matrix A to the identity, I, you multiply it by the inverse A^{-1}.

     A A^{-1} = A^{-1} A = I

    So the question is really, "Find the inverse of L and of U."
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  3. #3
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    Re: Find unkown value for matrices

    Quote Originally Posted by learncsmath View Post
    I am new at matrices and algebra. Kindly help me out to solve the problem from the first chapter of my textbook.

    For a lower triangular matrix L and an upper triangular matrix U of order three with unit diagonal elements determine P such that
    (a) LP = PL = I
    (b) UP = PU = I
    Are you given specific matrices L and U? This seems a very general problem which is going to have a very complicated solution if the off diagonal elements of L and U are "generic". Further, as Fernando Revilla says, you are really asked for a matrix that is the inverse of both L and U. For general L and U, that will not exist.
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  4. #4
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    Re: Find unkown value for matrices

    Actually, this problem is *not* that hard, provided it is two separate problems. The second one, for example, is to find U-1, where:

    U = \begin{bmatrix}1&a&b\\0&1&c\\0&0&1 \end{bmatrix}.

    you should know (hopefully) that the product of two upper-triangular matrices with diagonal entries of 1, is another such matrix. since I (the identity matrix) is also upper triangular, with diagonal entries all 1, it seems reasonable to conjecture that U-1 is also upper-triangular with 1's on the diagonal. so we want to solve THIS equation:

    \begin{bmatrix}1&a&b\\0&1&c\\0&0&1 \end{bmatrix} \begin{bmatrix}1&x&y\\0&1&z\\0&0&1 \end{bmatrix} = \begin{bmatrix}1&0&0\\0&1&0\\0&0&1 \end{bmatrix}

    for x,y and z, in terms of a,b and c. you can do this.
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  5. #5
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    Re: Find unkown value for matrices

    Quote Originally Posted by Deveno View Post
    Actually, this problem is *not* that hard, provided it is two separate problems. The second one, for example, is to find U-1, where:

    U = \begin{bmatrix}1&a&b\\0&1&c\\0&0&1 \end{bmatrix}.

    you should know (hopefully) that the product of two upper-triangular matrices with diagonal entries of 1, is another such matrix. since I (the identity matrix) is also upper triangular, with diagonal entries all 1, it seems reasonable to conjecture that U-1 is also upper-triangular with 1's on the diagonal. so we want to solve THIS equation:

    \begin{bmatrix}1&a&b\\0&1&c\\0&0&1 \end{bmatrix} \begin{bmatrix}1&x&y\\0&1&z\\0&0&1 \end{bmatrix} = \begin{bmatrix}1&0&0\\0&1&0\\0&0&1 \end{bmatrix}

    for x,y and z, in terms of a,b and c. you can do this.
    Thanks for the reply. I have solved the problem. As a beginner I was not aware that the product of two upper triangular matrices is also an upper triangular matrix. Keeping that concept in mind I solved for P and got the correct answer. Thanks again.
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  6. #6
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    Re: Find unkown value for matrices

    Glad you got it right.
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