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Math Help - Prove induction

  1. #1
    Junior Member
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    Prove induction

    If A = a 0
    0 b

    then

    A^n = a^n 0
    0 b^n

    [IMG]file:///C:/Users/Owner/AppData/Local/Temp/moz-screenshot.jpg[/IMG]
    Please see attachment its hard to display the question in text. Thanks.
    Attached Thumbnails Attached Thumbnails Prove induction-math.jpg  
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by captainjapan View Post
    If A = a 0
    0 b

    then

    A^n = a^n 0
    0 b^n

    [IMG]file:///C:/Users/Owner/AppData/Local/Temp/moz-screenshot.jpg[/IMG]
    Please see attachment its hard to display the question in text. Thanks.
     <br />
A^1=A=\left[\begin{array}{cc}<br />
a&0\\0&b<br />
\end{array}\right]<br />

    which will give us a base case.

    Suppose true for some k>1

    Then

     <br />
A^{k+1}=AA^k=\left[\begin{array}{cc}a&0\\0&b\end{array}\right]\left[\begin{array}{cc}a^k&0\\0&b^k\end{array}\right]<br />

    Now do the multiplication abd you should be there.

    CB
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  3. #3
    Senior Member
    Joined
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    Quote Originally Posted by captainjapan View Post
    If A = a 0
    0 b

    then

    A^n = a^n 0
    0 b^n

    [IMG]file:///C:/Users/Owner/AppData/Local/Temp/moz-screenshot.jpg[/IMG]
    Please see attachment its hard to display the question in text. Thanks.
    Have you ever heard about mathematical induction?

    n = 1: OK

    n -> n+1

     A^{n+1} = A^n \cdot A =  \begin{pmatrix} a^n & 0 \\ 0 & b^n \end{pmatrix}\cdot \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}

     = \begin{pmatrix} a^n \cdot a + 0 & 0 \\ 0 & b^n \cdot b \end{pmatrix}

     = \begin{pmatrix} a^{n+1} & 0 \\ 0 & b^{n+1} \end{pmatrix}

    Edit: Too slow
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