# matrixes

• Mar 4th 2010, 03:05 AM
vuze88
matrixes
I've got two questions which im struggling with:

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

2. Let $A$ be some $2\times2$ matrix such that $AX=XA$ for all real $2\times2$ matrxes. Prove that $A=\alphaI$ for $\alpha \in R$
• Mar 4th 2010, 04:16 AM
HallsofIvy
Quote:

Originally Posted by vuze88
I've got two questions which im struggling with:

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

Use $A^2= I$ and $B^2= I$ to show that (AB)(BA)= I. Then you have (AB)(BA)= (AB)(AB). Multiply both sides by $(AB)^{-1}$ (after showing that AB is invertible, of course).

Quote:

2. Let $A$ be some $2\times2$ matrix such that $AX=XA$ for all real $2\times2$ matrxes. Prove that $A=\alpha I$ for $\alpha \in R$
Let X be $\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}$, $\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}$, $\begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}$, and $\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}$.
• Mar 4th 2010, 05:33 AM
satx
The word is "matrices" not "matrixes" (Happy)
• Mar 4th 2010, 02:00 PM
vuze88
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"
• Mar 5th 2010, 04:12 AM
HallsofIvy
A matrix is invertible if and only if its determinant is non-zero. If A had determinant 0, then $det(A^2)= det(A)det(A)= 0$. but $A^2= I$ 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:
$\begin{barray}a & b \\ c & d\end{bmatrix}= a\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}+ b\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}+ c\begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}+ d\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}$
so they "represent" all 2 by 2 matrices.