Dividing Matrices

Matrix division is not formally defined mathematically. However, one can show an analogy between scalar division and matrix inversion. To introduce the concept of matrix division, consider the following linear scalar equation:

where A and b are known and x is unknown. Solving for x yields

where A^-1 denotes division and A is a non-zero value. Now consider a system of linear algebraic equations as follows

where A is a known square matrix, b is a known column vector, and x is a column vector containing n number of unknowns. Solving for the unknowns x, in the case of A is non-singular one can symbolically write:

where A^-1 is the inverse of matrix A. The matrix inverse A^-1 is analogous to matrix division A^-1 for the scalar equation:

Since the matrix multiplication of A^-1b must be a (n x 1) matrix.

If A^-1 post-multiplied b, then:

is invalid from a matrix multiplication standpoint because the inner indices are not equal, i.e., the number of columns of b is not equal to the number of rows of A^-1. One should note that A^-1 always pre-multiplies b,

In contrast for the scalar equation

where the order does not matter.