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Matrix Division

Tool to calculate matrix divisions of 2 matrices (2x2, 3x3, 4x4, 5x5, ...). The matrix division consists of the multiplication by an inverted matrix.

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Matrix Division -

Tag(s) : Matrix

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# Matrix Division

## Matrix Division

### How to make a division with matrices?

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 $n \times n$.

### How to divide a matrix by a scalar?

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

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