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Math Help - finding matrix A

  1. #1
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    finding matrix A

    Let A \in M_{2}(\mathbb(R)).
    If AB=BA for all B \in M_{2} (\mathbb(R)),
    show that A=\alpha I_{2} for any \alpha \in \mathbb(R).
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  2. #2
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    Quote Originally Posted by deniselim17 View Post
    Let A \in M_{2}(\mathbb(R)).
    If AB=BA for all B \in M_{2} (\mathbb(R)),
    show that A=\alpha I_{2} for any \alpha \in \mathbb(R).
    One way is to write A=\left[\begin{matrix}a&b\\c&d\end{matrix}\right] then select B=\left[\begin{matrix}1&1\\1&1\end{matrix}\right] which gives you a=d and b=c, then select B=\left[\begin{matrix}1&0\\0&0\end{matrix}\right] which gives you c=b=0. (Find AB and BA and set corresponding cells equal.)
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    Quote Originally Posted by undefined View Post
    One way is to write A=\left[\begin{matrix}a&b\\c&d\end{matrix}\right] then select B=\left[\begin{matrix}1&1\\1&1\end{matrix}\right] which gives you a=d and b=c, then select B=\left[\begin{matrix}1&0\\0&0\end{matrix}\right] which gives you c=b=0. (Find AB and BA and set corresponding cells equal.)
    Since the question asked "for all B", I let B=\left [\begin{matrix} w & x \\ y & z \end{matrix} \right].
    But I can't continue.
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    Quote Originally Posted by deniselim17 View Post
    Since the question asked "for all B", I let B=\left [\begin{matrix} w & x \\ y & z \end{matrix} \right].
    But I can't continue.
    If it's true for all B, it's certainly true for the particular B's I mentioned in my first post. Your approach might work but it seems a bit of a mess. Note that you're only asked to prove implication in one direction. Also note that the problem statement is a little imprecise in language, I would write "for some alpha" rather than "for any alpha". (Clearly it's an existential rather than universal statement, because the universal statement makes no sense.)
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