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  1. If two matrices $A$ and $B$ are commutative then all rules for real numbers $a$ and $b$ apply for the matrices?

    For example, if $AB=BA$ then:
    $(A+B)^2=A^2 + 2AB + B^2$
    $A^3 - B^3 = (A-B)(A^2+AB+B^2)$
    and so on...

  2. If the matrix $A$ is invertible then is $A^m A^n = A^{(m+n)}$, where $m,n$ are integers?

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    (1) The evaluation map $\Bbb C[x,y]\to \Bbb C[A,B]$ will be a ring homomorphism if $A,B$ commute. (I think this is the correct algebraic language for what you speak of.) (2) This holds very generally, not just for $A$ invertible (ie arbitrary monoids).2012-08-12
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    @anon: note that $m$ and/or $n$ might be negative.2012-08-12
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    [Oh, right; then in arbitrary groups.]2012-08-12
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    yes, you can show it by direct computation for the first identity.2012-08-12

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