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Math Help - Problem with a matrix

  1. #1
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    Exclamation Problem with a matrix

    Hi i got stuck on this problem - pls help if u can:

    Find all scalar matrices

    A^3 + 2A^2 - 4A - 8I_2 = 0

    where I_2 is identity matrix.
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  2. #2
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    Quote Originally Posted by karagorge View Post
    Hi i got stuck on this problem - pls help if u can:

    Find all scalar matrices

    A^3 + 2A^2 - 4A - 8I_2 = 0

    where I_2 is identity matrix.
    Given the above get the polynomial

    x^3+2x^2-4x-8=0 \iff x^2(x+2)-4(x+2)=0 \iff (x+2)^2(x-2)=0

    Since this polynomial is in the annihilating Ideal the minimum polynomial must divide the above polynomial

    Since we are working with

    2 \times 2

    Matrices
    We can have the following minimum polynomials
    (x+2);\quad (x+2)^2; \quad (x-2); \quad (x+2)(x-2)

    So up to similarity you will get the 4 matrices

    \begin{bmatrix} -2 & 0 \\ 0 & -2\end{bmatrix}

    \begin{bmatrix} -2 & 1 \\ 0 & -2\end{bmatrix}

    \begin{bmatrix} 2 & 0 \\ 0 & 2\end{bmatrix}

    \begin{bmatrix} -2 & 0 \\ 0 & 2\end{bmatrix}
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