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Matrix Division

Tool to calculate matrix algebra divisions. The matrix division consists of the multiplication by an inverted matrix.

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# Matrix Division

## Matrix Division

### Division of a Matrix by a Scalar (Number)

Tool to calculate matrix algebra divisions. The matrix division consists of the multiplication by an inverted matrix.

### How to make a division with matrices?

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 $$M_2$$ : $$M_2^{-1}$$.

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

$$\begin{bmatrix} 0 & 1 \\ 2 & 3 \end{bmatrix} / \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 2 & 3 \end{bmatrix} . \left( \frac{1}{2} \begin{bmatrix} -4 & 2 \\ 3 & -1 \end{bmatrix} \right) = \frac{1}{2} \begin{bmatrix} 3 & -1 \\ 1 & 1 \end{bmatrix}$$

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 invertible, the matrix $$M_2$$ must be 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 the initial matrix $$M$$, with each items of the matrix divided by $$\lambda$$.

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

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