I am asked to show that the set of matrices
$$G=\left\{\begin{bmatrix}1&a&b\\0&1&c\\0&0&1\end{bmatrix}:a,b,c\in\mathbb Q\right\}$$
form an abelian group wrt matrix multiplication. (Assume that matrix multiplication is associative).
I know that for $G$ to form an abelian group under matrix multiplication,
- Matrix multiplication in $G$ should be associative.
- Existence of identity element in matrix multiplication.
- Existence of inverse element in matrix multiplication.
- Matrix multiplication in $G$ should be commutative.
For $1$, it is already given that matrix multiplication is associative.
For $2$, I have found the identity element $a = b = c = 0$.
For $3$, I have similarly found the inverse element.
But for $4$, I am unable to prove that $AB = BA$ for all $A,B\in G$ containing terms a1,b1,c1,a2,b2,c2 since a1c2 is not equal to a2c1.
How do show that $G$ is an abelian group under matrix multiplication? Please help.
Best Answer
You can't prove commutativity because it's false. Let $$A=\begin{bmatrix}1&1&0\\0&1&0\\0&0&1\end{bmatrix}\qquad B=\begin{bmatrix}1&0&0\\0&1&1\\0&0&1\end{bmatrix}$$ Then $$AB=\begin{bmatrix}1&1&1\\0&1&1\\0&0&1\end{bmatrix}\ne\begin{bmatrix}1&1&0\\0&1&1\\0&0&1\end{bmatrix}=BA$$ $G$ is still a group under matrix multiplication, but not an abelian group.