Matrix Division

A matrix $ M_1 $ of $ m $ lines and $ n $ columns and $ M_2 $ a square matrix of $ n \times n $. The dividing matrices operation $ M_1/M_2 $ consist in the multiplication of the matrix $ M_1 $ by the inverse matrix of $ M_2 $ : $ M_2^{-1} $.

$$ M_1/M_2 = M_1 \times M_2^{-1} $$

Example: Division of 2x2 matrices $$ \begin{bmatrix} 0 & 1 \\ 2 & 3 \end{bmatrix} / \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 2 & 3 \end{bmatrix} . \left( \frac{1}{2} \begin{bmatrix} -4 & 2 \\ 3 & -1 \end{bmatrix} \right) = \frac{1}{2} \begin{bmatrix} 3 & -1 \\ 1 & 1 \end{bmatrix} $$

To make the division, the multiplication">matrix multiplication rules must be followed: $ M_1 $ must have the same number $ n $ of columns as the number of rows of $ M_2 $. Moreover, to be an invertible matrix, the $ M_2 $ matrix must be a square and therefore of size $ n \times n $.

The division of the matrix $ M=[a_{ij}] $ by a scalar $ \lambda $ is a matrix of the same size than $ M $ (the initial matrix), with each items of the matrix divided by $ \lambda $.

$$ \frac{M}{\lambda} = [ a_{ij} / \lambda ] $$

Example: $$ \begin{bmatrix} 0 & 2 \\ 4 & 6 \end{bmatrix} / 2 = \begin{bmatrix} 0 & 1 \\ 2 & 3 \end{bmatrix} $$