Is division of matrices possible? Is it possible to divide a matrix by another? If...

Shirley Thompson

Shirley Thompson

Answered

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 AB if
A=(abcd)

B=(wxyz)?

Answer & Explanation

Marcus Herman

Marcus Herman

Expert

2021-12-17Added 41 answers

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, xy is really the same as multiplying x and y1=1y. We call y1 the inverse of y, and note that it has the property that y1=1.
The same goes for different algebraic structures. That is, for two elements x, y in this algebraic structure we define xy as xy1 (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
A1=1(adbc)[dbca]
and you should check that AA1=E, the identity matrix. Now, for two matrices B and A, BA=BA1

Chanell Sanborn

Chanell Sanborn

Expert

2021-12-18Added 41 answers

For ordinary numbers ab 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 AB1orB1A, 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

Jeffrey Jordon

Expert

2021-12-26Added 2575 answers

Normally, matrix division is defined as AB=AB1 where B1 stands for the inverse matrix of B. In the case where the inverse doesn't exist the so called pseudoinverse may be used.

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