matrices why can't they be divided?

Question

Hi, I would like to know how come or why you can't divide a matrix by another matrix
thanks for any help received

i was looking more for a paragraph on why it can't be done

Answer 1

Well you can, in a sense.
I don't know a SIMPLE way to do it, in the sense that there is a SIMPLE way to do matrix multiplication. But maybe the following explanation will be helpful:

If [A] x [B] = [C] (i.e., if the product of matrices A and B equals matrix C), then we can say that [C] / [B] = [A]. So in that sense, we are dividing C by B.

But what if we don't know matrix A. What if we know only [B] and [C]? How can we divide [C] by [B] to find [A]?

The answer is that we find the "inverse" of matrix B, and then MULTIPLY matrix C by this inverse matrix (which is shown as [B]^-1). We define [B]^-1 as the matrix for which [B] x [B]^-1 = the identity matrix. (The identity matrix is a square matrix that has 1's on its main diagonal and 0's everywhere else.)

The problem, then, becomes how to find the inverse of [B]. Fortunately, there are procedures for this, and you can probably find them on line using Google. Not all matrices can be inverted. But if B can be inverted, then you have:

[C] / [B] = [C] x [B]^-1 = [A]

Note that if we substitute [A] x [B] for [C], then this becomes;
([A] x [B]) / [B] = [A] x [B] x [B]^-1 = [A] x identity matrix = [A]

Answer 2

Everything that can be done with matrices can be done without them. Matrices are essentially shorthand and matrix multiplication is something which we have defined. We have not defined matrix division as such.

Division in the real numbers is the same as multiplying by a multiplicative invers. 1/2 is the inverse of 2. When you multiply 2 and 1/2, are you multiplying or dividing. That is up to the person who makes definitions. If these numbers were to be defined like matrices are, we would say that we are multiplying them and that there is no such thing as division in our system. Every number in the real number system except zero has an inverse, so we would call zero non-invertible.

An invertible matrix is, by definition, one which can be multiplied by another matrix in your system to give you the identity matrix. Lots of matrices are non-invertible, so we would run into lots of problems if we tried to divide by these. It would be like dividing by many different flavors of zero. We can get around this by multiplying inverse matrices instead of dividing by non inverted matrices.

That's the long answer.

The short answer is this: by definition. We have not defined matrix division, so we cannot use it.