Show that subtraction of matrices is neither commutative nor associative.
My Work
Let $A$ and $B$ be $m \times n$ matrices with elements $a_{ij}$ and $b_{i j}$, respectively.
Commutativity says that $A + B = B + A$.
$A - B = a_{i,j} - b_{i,j}$
$B - A = b_{i,j} - a_{i,j}$
Since subtraction of numbers is not commutative, we have that $a_{i,j} - b_{i,j} \not= b_{i,j} - a_{i,j}$.
$\therefore A - B \not= B - A$.
Let $A, B, $ and $C$ be $m \times n$ matrices with elements $a_{ij}$, $b_{i j}$, and $c_{i,j}$ respectively.
$(A - B) - C = (a_{i,j} - b_{i,j}) - c_{i,j}$
$A - (B - C) = a_{i,j} - (b_{i,j} - c_{i,j}) = a_{i,j} - b_{i,j} + c_{i,j} = (a_{i,j} - b_{i,j}) + c_{i,j} \not= (a_{i,j} - b_{i,j}) - c_{i,j}$
I would greatly appreciate it if people could please take the time to review my work for correctness and provide feedback.