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Math Help - Matrices

  1. #1
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    Matrices

    Hi I'm not sure if this is the right forum to post this, but if it is not, I sincerely apologize.

    I am really stuck at these two questions,

    1.) Find all 2x2 matrices A for which A^2=A by letting A = \begin{pmatrix} a & b \\c & d \end{pmatrix}

    This was the last part to this question, "The result "If ab=0 then a=0 or b=0 for real numbers does not have an equivalent result for matrices."
    I don't know if this could be helpful but I really do not know how to do this question..

    2.) Give one example which shows that if "A^2=O then A=O" is a false statement. (O is the null matrix!)

    I got up to multiplying the matrices together and then equated it to 0..


    Really appreciate your help!
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  2. #2
    Junior Member Barioth's Avatar
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    Re: Matrices

    if A= \begin{pmatrix} a & b \\ c & d \end{pmatrix}
    then A^2= \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix}=\begin{pmatrix} a^2+bc & ab+bd \\ ca+dc & cb+d^2 \end{pmatrix}

    so in 1) you have the four equation

    \\a= a^2+bc\\b= ab+bd\\c=ca+dc\\ d = cb+d^2

    So now you have to solve this system.

    and as of for 2)

    \\0= a^2+bc\\0= ab+bd\\0=ca+dc\\ 0 = cb+d^2

    you must solve this system.
    I gave an exemple (it is not unique) of a solution, but you should try to find it before checking.
    Spoiler:

    \begin{pmatrix} \pi & -\pi \\ \pi & -\pi \end{pmatrix}
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