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Math Help - Matrix

  1. #1
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    Matrix

    Prove that if \mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A} then (\mathbf{A}+\mathbf{B})^n=\binom n0\mathbf{A}^n+\binom n1\mathbf{A}^{n-1}\mathbf{B}+\dots + \binom n{n-1} \mathbf{A}\mathbf{B}^{n-1}+\binom nn\mathbf{B}^n.
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  2. #2
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    Quote Originally Posted by james_bond View Post
    Prove that if \mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A} then (\mathbf{A}+\mathbf{B})^n=\binom n0\mathbf{A}^n+\binom n1\mathbf{A}^{n-1}\mathbf{B}+\dots + \binom n{n-1} \mathbf{A}\mathbf{B}^{n-1}+\binom nn\mathbf{B}^n.
    Hi

    You can first prove by induction that
    \mathbf{B}^k\mathbf{A}=\mathbf{A}\mathbf{B}^k

    And use this to prove by induction
    \left(\mathbf{A}+\mathbf{B}\right)^n=\sum_{k=0}^{n  } \binom nk\mathbf{A}^k\mathbf{B}^{n-k}

    \left(\mathbf{A}+\mathbf{B}\right)^{n+1}=\left(\su  m_{k=0}^{n} \binom nk\mathbf{A}^k\mathbf{B}^{n-k}\right)\left(\mathbf{A}+\mathbf{B}\right)

    \left(\mathbf{A}+\mathbf{B}\right)^{n+1}=\sum_{k=0  }^{n} \binom nk\mathbf{A}^k\mathbf{B}^{n-k}\mathbf{A}+\sum_{k=0}^{n} \binom nk\mathbf{A}^k\mathbf{B}^{n+1-k}

    \left(\mathbf{A}+\mathbf{B}\right)^{n+1}=\sum_{k=0  }^{n} \binom nk\mathbf{A}^{k+1}\mathbf{B}^{n-k}+\sum_{k=0}^{n} \binom nk\mathbf{A}^k\mathbf{B}^{n+1-k}
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