Why a matrix cannot be divided?

Question

We know that we can operate addition$(+)$ , subtraction$(-)$ , multiplication$(×)$ on a matrix. But why we can't operate division $(÷)$ on a matrix?...

Sorry for my bad english

Answer 1

First, you should note that although the symbols $+,-,×$ as used in matrix arithmetic are similar to those used in the arithmetic of integers, they have slightly different meanings. That is, the corresponding operations on matrixes are defined differently, and there are some special additional constraints. For example, we never consider the class of all possible matrices, but only define addition (this includes subtraction) on matrices of equal order. The way multiplication of matrices is defined really gives them more structure, but we also have severe restrictions on when this operation is defined. In general we can think of the set of all $n×n$ matrices (for some fixed $n$) as an appropriate algebraic structure depending on what we want to use them for.

Now, consider one such set, say when $n=3.$ Let us call this set $M_3.$ If we've already checked that although $+$ in $M_3$ makes it into an abelian group (so that we can subtract matrices in $M_3$ and not leave the space), we may soon notice that in general $AB\ne BA$ whenever $A,B \in M_3.$ However, that is not so disappointing. But when we try to see if for all $A\in M_3$ there is some $B$ so that $AB=I_3,$ where $I_3$ is the identity matrix wrt $×$ in $M_3$ (for this is what will allow us to "divide" in this space), we discover that we cannot in general pull this off (even apart from the case of the null matrix $O_3$). This disappoints us, but what can we do about it?