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Math Help - Matrix Proof

  1. #1
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    Matrix Proof

    I need to prove that these determinants are equals.
    Note: these are determinants, not matrices, I just don't know how to set them up as determinants using the math code...

    \begin{bmatrix}a1+b1t&a2+b2t&a3+b3t\\a1t+b1&a2t+b2  &a3t+b3\\c1&c2&c3 \end{bmatrix} = (1- t^2) \begin{bmatrix}a1&a2&a3\\b1&b2&b3\\c1&c2&c3 \end{bmatrix}

    This tells me that I need to somehow get (1- t^2) multiplied into the determinant. I know that to get that out, one would have to multiply a row by (1- t^2), or possibly (1-t) on two different rows... Am I thinking it the right direction?
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  2. #2
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    Quote Originally Posted by Hellreaver View Post
    I need to prove that these determinants are equals.
    Note: these are determinants, not matrices, I just don't know how to set them up as determinants using the math code...

    \begin{bmatrix}a1+b1t&a2+b2t&a3+b3t\\a1t+b1&a2t+b2  &a3t+b3\\c1&c2&c3 \end{bmatrix} = (1- t^2) \begin{bmatrix}a1&a2&a3\\b1&b2&b3\\c1&c2&c3 \end{bmatrix}

    This tells me that I need to somehow get (1- t^2) multiplied into the determinant. I know that to get that out, one would have to multiply a row by (1- t^2), or possibly (1-t) on two different rows... Am I thinking it the right direction?
    Consider the matrix A = \begin{bmatrix}a1&a2&a3\\b1&b2&b3\\c1&c2&c3 \end{bmatrix}.

    Apply the row operation R1 --> R1 + t R2 on matrix A to get matrix B: det(B) = det(A).

    Apply the row operation (1 - t^2) R2 + R1 --> R2 on matrix B to get matrix C: det(C) = (1 - t^2) det(B) ......
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Consider the matrix A = \begin{bmatrix}a1&a2&a3\\b1&b2&b3\\c1&c2&c3 \end{bmatrix}.

    Apply the row operation R1 --> R1 + t R2 on matrix A to get matrix B: det(B) = det(A).

    Apply the row operation (1 - t^2) R2 + R1 --> R2 on matrix B to get matrix C: det(C) = (1 - t^2) det(B) ......
    Ok, that's tricky. Thank you so much!
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