 Shirley Thompson

2021-12-16

Is division of matrices possible?
Is it possible to divide a matrix by another? If yes, What will be the result of $\frac{A}{B}$ if
$A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$

$B=\left(\begin{array}{cc}w& x\\ y& z\end{array}\right)?$ Marcus Herman

Expert

There is a way to performa sort of division , but I am not sure if it is the way you are looking for. For motivation ,consider the ordinary real numbers R . We have that for two real numbers, $\frac{x}{y}$ is really the same as multiplying x and ${y}^{-1}=\frac{1}{y}.$ We call ${y}^{-1}$ the inverse of y, and note that it has the property that $y-1=1.$
The same goes for different algebraic structures. That is, for two elements x, y in this algebraic structure we define $\frac{x}{y}$ as $x{y}^{-1}$ (under some operation). Most notably, we have a notion of division in any division ring (hence the name!) . It turns out that if you consider invertible $n×n$ matrices with addition and ordinary matrix multiplication, there is a sensible way to define division since every invertible matrix has well, an inverse. So just to help you grip what an inverse is, say that you have a $2×2$ matrix
The inverse of A is then given by
${A}^{-1}=\frac{1}{\left(ad-bc\right)}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$
and you should check that $A{A}^{-1}=E$, the identity matrix. Now, for two matrices B and A, $\frac{B}{A}=B{A}^{-1}$ Chanell Sanborn

Expert

For ordinary numbers $\frac{a}{b}$ means the solution to the equation $xb=a$. This is the same as $bx=a$, but since matrix multiplication is not commutative, there are two different possible generalizations of "division" to matrices.
If B is invertible, then you can form $A{B}^{-1}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}{B}^{-1}A$, but these are not in general the same matrix. They are the solutions to $XB=A$ and $BX=A$ respectively.
If B is not invertible, then $XB=A$ and $BX=A$ may have solutions, but the solutions will not be unique. So in that situation speaking of "matrix division" is even less warranted. Jeffrey Jordon

Expert

Normally, matrix division is defined as $\frac{A}{B}=A{B}^{-1}$ where ${B}^{-1}$ stands for the inverse matrix of B. In the case where the inverse doesn't exist the so called pseudoinverse may be used.