Matrix Division

Consider \( M_1 \) a matrix of \( m \) lines and \( n \) columns and \( M_2 \) a square matrix of \( n \times n \). The matrix division \( M_1/M_2 \) 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: $$ \begin{bmatrix} 0 & 1 \\ 2 & 3 \end{bmatrix} / \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 2 & 3 \end{bmatrix} . \left( \frac{1}{2} \begin{bmatrix} -4 & 2 \\ 3 & -1 \end{bmatrix} \right) = \frac{1}{2} \begin{bmatrix} 3 & -1 \\ 1 & 1 \end{bmatrix} $$

To make the division, the 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 the initial matrix \( M \), with each items of the matrix divided by \( \lambda \).

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

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