Is Matrix Multiplication Commutative? I am doing some self-study in math. Below is a problem that I did but I am not sure if I have the right answer. I am hoping that somebody could check it for me.
Thanks
Bob
Problem:
If $A$ and $B$ are two by two matrices, does it imply that $A * B = B * A$?
Answer:
\begin{eqnarray*}
A &=& \begin{bmatrix}
a_{1\,1} & a_{1\,2} \\
a_{2\,1} & a_{2\,2} \\
\end{bmatrix} \\
B &=& \begin{bmatrix}
 b_{1\,1} & b_{1\,2} \\
 b_{2\,1} & b_{2\,2} \\
\end{bmatrix} \\
A * B &=& \begin{bmatrix}
 a_{1\,1} b_{1 \, 1} + a_{1 \, 2 }b_{2 \, 1} &
  a_{1\,1} b_{1\,2} + a_{1\,2}b_{2\,2 } \\
 a_{2\,1} b_{1\,1} + a_{2\,2}b_{2\,1} & a_{2\,1} b_{1\,2} + a_{2\,2}b_{2\,2} \\
\end{bmatrix} \\
B *A &=& \begin{bmatrix}
 b_{1\,1} a_{1 \, 1} + b_{1 \, 2 }a_{2 \, 1} &
 b_{1\,1} a_{1\,2} + b_{1\,2}a_{2\,2 } \\
 b_{2\,1} a_{1\,1} + b_{2\,2}a_{2\,1} & b_{2\,1} a_{1\,2} + b_{2\,2}a_{2\,2} \\
\end{bmatrix} \\
\end{eqnarray*}
Assume that $A * B = B * A$ then we have:
\begin{eqnarray*}
a_{1\,1} b_{1 \, 1} + a_{1 \, 2 }b_{2 \, 1} &=&
 b_{1\,1} a_{1 \, 1} + b_{1 \, 2 }a_{2 \, 1} \\
\end{eqnarray*}
If $a_{1\,1} = b_{1 \, 1} = 1$ we have:
\begin{eqnarray*}
1(1) + a_{1 \, 2 }b_{2 \, 1} &=& 1(1) + b_{1 \, 2 }a_{2 \, 1} \\
a_{1\,2} b_{2\,1} &=& b_{1\,2}a_{2\,1} \\
\end{eqnarray*}
If $a_{1\,2} = b_{2 \, 1} = b_{1\,2} = 1$ and $a_{2\,1} = 0$ we have:
\begin{eqnarray*}
1(1) &=& 1(0) \\
1 &=& 0 \\
\end{eqnarray*}
Therefore, I conclude for the set of two by two matrices the operator $*$ is
not communicative.

Your approach is correct, but there is an easier way:
You can prove that something is not true by giving a counterexample (the claim is that matrix multiplication is not commutative, so it is sufficient to find at least one example where commutativity fails). So, a much easier approach is:
$$\begin{pmatrix} 1 &0\\0&0\end{pmatrix} \begin{pmatrix} 1 &0\\1&0\end{pmatrix} = \begin{pmatrix} 1 &0\\0&0\end{pmatrix}$$
$$ \begin{pmatrix} 1 &0\\1&0\end{pmatrix}\begin{pmatrix} 1 &0\\0&0\end{pmatrix} = \begin{pmatrix} 1 &0\\1&0\end{pmatrix}$$
Hence, matrix multiplication is not commutative.
You can ask yourself why matrix multiplication is defined this way. This definition seems somewhat strange when one is exposed to matrix multiplication for the first time. The deeper underlying reason is that we can represent a certain type of function, called linear transformation, with matrices. Matrix multiplication is defined such that it corresponds with function composition of linear transformation. Since function composition is not commutative, it also makes sense that matrix multiplication is not commutative. 
