1. ## 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 $\displaystyle \alpha \in R$

2. Originally Posted by vuze88
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 $\displaystyle \begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}$.

3. The word is "matrices" not "matrixes"

4. 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"

5. 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.