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Math Help - Help with matrices

  1. #1
    Junior Member
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    Help with matrices

    Let A denote the 2x2 matrix with a_{11} = 2, a_{12} = 0, a_{21} = 4 and a_{22} = 1. Find a polynomial p(x) with degree two such that p(A) = 0

    How can I find this one?

    And how can I prove that  (A+B)^2 = A^2 + 2AB + B^2 and (A-B)(A+B) = A^2 - B^2, if A and B are square matrices of the same size.

    Thank you.
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  2. #2
    MHF Contributor red_dog's Avatar
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    Use the Cayley-Hamilton identity:

    A^2-Tr(A)\cdot A+\det(A)\cdot I_2=O_2

    where Tr(A)=a_{11}+a_{22}, \ \det(A)=a_{11}a_{22}-a_{12}a_{21}

    and I_2=\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}, \ O_2=\begin{pmatrix}0 & 0\\0 & 0\end{pmatrix}
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