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

Matrix Division - dCode

Tag(s) : Matrix

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

To calculate a matrix division, proceed in stages:

— Check that the number of columns of the matrix $ M_1 $ is equal to the number of rows of the matrix $ M_2 $

— Check that the matrix $ M_2 $ is a square matrix (same number of rows and columns: 2x2, 3x3, 4x4, NxN).

— Check that the matrix $ M_2 $ is an invertible matrix.

— Calculate the inverse of the matrix $ M_2 $ noted $ M_2^{-1} $

— Calculate the multiplication">matrix multiplication $ M_1 \times M_2^{-1} $$ the result obtained is that of the matrix division.

__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 as $ M $ (the initial matrix), with each item 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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Cite as source (bibliography):

*Matrix Division* on dCode.fr [online website], retrieved on 2023-10-01,

division,matrix,scalar,inverse,2x2,3x3,4x4

https://www.dcode.fr/matrix-division

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