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Math Help - MAtrix Problem

  1. #1
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    MAtrix Problem

    If P=\begin{bmatrix} 1 & -1 \\2 & -1 \end{bmatrix}, Q=\begin{bmatrix} 1 & 1 \\ q & -1 \end{bmatrix}

    (P+Q)^2 = P^2 + Q^2 , determine the value of q
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  2. #2
    MHF Contributor
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    Hi

    P+Q=\begin{bmatrix} 2 & 0 \\2+q & -2 \end{bmatrix}
    (P+Q)^2=\begin{bmatrix} 4 & 0 \\0 & 4 \end{bmatrix}

    P^2=\begin{bmatrix} -1 & 0 \\0 & -1 \end{bmatrix}

    Q^2=\begin{bmatrix} 1+q & 0 \\0 & 1+q \end{bmatrix}
    P^2 + Q^2=\begin{bmatrix} q & 0 \\0 & q \end{bmatrix}

    (P+Q)^2 = P^2 + Q^2 for q=4
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  3. #3
    Super Member flyingsquirrel's Avatar
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    Hi,

    A slightly different method which might save you some calculations :

    (P+Q)^2=P^2+PQ+QP+Q^2 hence (P+Q)^2=P^2+Q^2 \Longleftrightarrow PQ+QP=0\quad (1). Now all you have to do is to compute PQ+QP and solve (1) for q.
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