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Math Help - Number Theory Matrix Help

  1. #1
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    Number Theory Matrix Help

    Suppose that a matrix A =
    a 0

    0 b

    where a and b are real numbers

    Prove that A^n=
    a^n 0
    0 b^n

    for every positive integer n
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  2. #2
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    Matrix proof

    Hello tokio
    Quote Originally Posted by tokio View Post
    Suppose that a matrix A =
    a 0

    0 b

    where a and b are real numbers

    Prove that A^n=
    a^n 0
    0 b^n

    for every positive integer n
    Use induction: it takes one line:

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

    I'll leave the rest to you.

    Grandad
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  3. #3
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    Quote Originally Posted by Grandad View Post
    Hello tokioUse induction: it takes one line:

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

    I'll leave the rest to you.

    Grandad
    I'm confused on how to set this up to do the induction , im trying the hardest to understand, things are just not clicking right now.
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  4. #4
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    Proof by induction

    Hello Tokio

    Given that A = \begin{pmatrix}a & 0\\0 & b\end{pmatrix}, let P(n) be the propositional function: A^n =\begin{pmatrix}a^n & 0\\0 & b^n\end{pmatrix}

    Then P(n) \Rightarrow A^{n+1}= \begin{pmatrix}a^n & 0\\0 & b^n\end{pmatrix}\begin{pmatrix}a & 0\\0 & b\end{pmatrix} = \begin{pmatrix}a^{n+1} & 0\\0 & b^{n+1}\end{pmatrix}\Rightarrow P(n+1)


    P(1) is true. So A^n =\begin{pmatrix}a^n & 0\\0 & b^n\end{pmatrix}, \forall n \in \mathbb{N}

    Grandad
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