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Math Help - Solving a Matrix using its Determinant

  1. #1
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    Question Solving a Matrix using its Determinant

    Hi all, I'm practicing for a mid-semester linear algebra exam and I'm stuck on this question:

    Let Ax=b where A = \begin{bmatrix} 2 & 1 & 0 \\ -3 & 0 & 1\\ 0 & 1 & 2 \end{bmatrix} , det A = 4 and b = \begin{bmatrix} 7 \\ -8 \\ -3 \end{bmatrix}. What does x2 equal?

    I know I could just use row reduction to solve for x2 but I would like to know how to use the determinant to make the problem easier.

    Thanks,
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  2. #2
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    Re: Solving a Matrix using its Determinant

    Quote Originally Posted by anguished View Post
    Hi all, I'm practicing for a mid-semester linear algebra exam and I'm stuck on this question:

    Let Ax=b where A = [2, 1, 0; -3, 0, 1; 0, 1, 2], det A = 4 and b = [7, -8, -3]. What does x2 equal?

    I know I could just use row reduction to solve for x2 but I would like to know how to use the determinant to make the problem easier.

    Thanks,
    Are you sure b doesn't equal [7; -8; -3]?
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    Re: Solving a Matrix using its Determinant

    Yes, that's what I meant sorry. I should have used Latex instead. :P Fixed now.
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    Re: Solving a Matrix using its Determinant

    Quote Originally Posted by anguished View Post
    Hi all, I'm practicing for a mid-semester linear algebra exam and I'm stuck on this question:

    Let Ax=b where A = \begin{bmatrix} 2 & 1 & 0 \\ -3 & 0 & 1\\ 0 & 1 & 2 \end{bmatrix} , det A = 4 and b = \begin{bmatrix} 7 \\ -8 \\ -3 \end{bmatrix}. What does x2 equal?

    I know I could just use row reduction to solve for x2 but I would like to know how to use the determinant to make the problem easier.

    Thanks,
    You need to realise that if \displaystyle \mathbf{A}\mathbf{x} = \mathbf{b}, then \displaystyle \mathbf{x} = \mathbf{A}^{-1}\mathbf{b}.

    You can use the determinant to help evaluate \displaystyle \mathbf{A}^{-1}.
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