Name: Muhammad S.
Date: Fall 2012
Matrices can be added, subtracted, and multiplied. Why is division not allowed?
It is, in a manner. Some square matrices can be inverted, so that the inverse matrix acts in many ways as a reciprocal.
Richard E. Barrans Jr., Ph.D., M.Ed.
Department of Physics and Astronomy
University of Wyoming
You CAN divide matrices. But only certain matrices apply and the
rules of operation are slightly different.
Firstly, matrices are not real numbers. It is a system of equations
that represent vectors and polynomial equations.
Secondly, division of real numbers is based on multiplication where
a * b is rewritten as a * (1/b) where (1/b) is called the inverse of
b. With the restriction that b <> 0.
In parallelism, dividing two matrices A/B is again A * Inverse
B. But the inverse of B implies that B is a square matrice AND its
determinant is non zero. Sound familiar?
So in summary, you can divide matrices using multiplication, but the
dividing matrice must be invertible, a square matrix and its
determinant must not equal zero.
With matrices, division is not definite. There are usually many different possible results to a division problem with matrices. A simple example is (4) divided by (1 1 1 1). I cannot type well in multiple dimensions, so please accept the last matrix as a column. Three possible answers are (4 0 0 0), (1 1 1 1), and (2 0 2 0). Addition, subtraction, and multiplication have definite answers.
Dr. Ken Mellendorf
Illinois Central College
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Update: November 2011