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Thread: matrixes

  1. #1
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    matrixes

    I've got two questions which im struggling with:

    1. Let $\displaystyle A$ and $\displaystyle B$ be $\displaystyle n\times n$ matrixes such that $\displaystyle A^2=I$,$\displaystyle B^2=I$ and $\displaystyle (AB)^2=I$. Prove that $\displaystyle AB=BA$

    2. Let $\displaystyle A$ be some $\displaystyle 2\times2$ matrix such that $\displaystyle AX=XA$ for all real $\displaystyle 2\times2$ matrxes. Prove that $\displaystyle A=\alphaI$ for $\displaystyle \alpha \in R$
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  2. #2
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    Quote Originally Posted by vuze88 View Post
    I've got two questions which im struggling with:

    1. Let $\displaystyle A$ and $\displaystyle B$ be $\displaystyle n\times n$ matrixes such that $\displaystyle A^2=I$,$\displaystyle B^2=I$ and $\displaystyle (AB)^2=I$. Prove that $\displaystyle AB=BA$
    Use $\displaystyle A^2= I$ and $\displaystyle B^2= I$ to show that (AB)(BA)= I. Then you have (AB)(BA)= (AB)(AB). Multiply both sides by $\displaystyle (AB)^{-1}$ (after showing that AB is invertible, of course).

    2. Let $\displaystyle A$ be some $\displaystyle 2\times2$ matrix such that $\displaystyle AX=XA$ for all real $\displaystyle 2\times2$ matrxes. Prove that $\displaystyle A=\alpha I$ for $\displaystyle \alpha \in R$
    Let X be $\displaystyle \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}$, $\displaystyle \begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}$, $\displaystyle \begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}$, and $\displaystyle \begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}$.
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  3. #3
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    The word is "matrices" not "matrixes"
    Last edited by satx; Mar 4th 2010 at 05:56 AM.
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  4. #4
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    for the first one, how do i prove AB is invertible? do i need to prove anythin else other than the fact that its square?

    for the second one, how did you know to choose those 4 "matrices"
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  5. #5
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    A matrix is invertible if and only if its determinant is non-zero. If A had determinant 0, then $\displaystyle det(A^2)= det(A)det(A)= 0$. but $\displaystyle A^2= I$ which has determinant 1, not 0. Therefore det(A) is not 0. Same thing for B. Then det(AB)= det(A)det(B) is non-zero.

    I chose those four matrices because they are the "standard basis" for the vector space of 2 by 2 matrices. Every such matrix can be written as a linear combination of them:
    $\displaystyle \begin{barray}a & b \\ c & d\end{bmatrix}= a\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}+ b\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}+ c\begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}+ d\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}$
    so they "represent" all 2 by 2 matrices.
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