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Math Help - URGENT...DUE TODAY

  1. #1
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    URGENT...DUE TODAY

    I'm a little confused about similar matrices.

    Are the the following matrices similar?

    A = \begin{bmatrix} 1 & 2 & 1 \\ 0 & -2 & 1 \\ 1 & 0 & 2 \end{bmatrix} \ \  \ B = \begin{bmatrix} 1 & 0 & 3\\ 1 & 0 & 1 \\ 2 & 1 & 1\end{bmatrix}

    Now I know that similar matrices share a variety of properties....some of which I've already calculated.

     For A \ \Rightarrow \ Rank = 2, Determinant = 0, Trace = 1, Char. Polynomial =  \lambda^3 - \lambda^2 - 5 \lambda

     For B \ \Rightarrow \ Rank = 3, Determinant = 2, Trace = 2, Char. Polynomial =  -2 + \lambda^3 - 2 \lambda^2 - 6 \lambda

    From this I would say that these two aren't similar, however, I have read other posts on this forum that make me think that the above properties aren't enough to safely conclude that the are or are not similar.

    I am aware that if 2 matrices, A and B, are similar then there is a matrix P such that  B = P^{-1} AP .

    How then, do I calculate P?

    Thanks in advance for any help
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  2. #2
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    Quote Originally Posted by Jimmy_W View Post
    I'm a little confused about similar matrices.

    Are the the following matrices similar?

    A = \begin{bmatrix} 1 & 2 & 1 \\ 0 & -2 & 1 \\ 1 & 0 & 2 \end{bmatrix} \ \  \ B = \begin{bmatrix} 1 & 0 & 3\\ 1 & 0 & 1 \\ 2 & 1 & 1\end{bmatrix}

    Now I know that similar matrices share a variety of properties....some of which I've already calculated.

     For A \ \Rightarrow \ Rank = 2, Determinant = 0, Trace = 1, Char. Polynomial =  \lambda^3 - \lambda^2 - 5 \lambda

     For B \ \Rightarrow \ Rank = 3, Determinant = 2, Trace = 2, Char. Polynomial =  -2 + \lambda^3 - 2 \lambda^2 - 6 \lambda

    From this I would say that these two aren't similar, however, I have read other posts on this forum that make me think that the above properties aren't enough to safely conclude that the are or are not similar.

    I am aware that if 2 matrices, A and B, are similar then there is a matrix P such that  B = P^{-1} AP .

    How then, do I calculate P?

    Thanks in advance for any help
    you don't need to find P. you've shown that \det A \neq \det B, which is enough to conclude that A and B are not similar.
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