Matrix division

The division of two matrices is the multiplication of a matrix by the inverse matrix of the dividing matrix, and at the same time, it requires that the dividing matrix be a square matrix and that its determinant be non-zero.

Matrix division

In other words, the division of two matrices is the multiplication of a matrix by the inverse matrix of the matrix that acts as a divisor and, as requirements of inverse matrices, they need to be square and the determinant to be non-zero.

It may seem contradictory that in order to divide two matrices we have to multiply them. The key is that in this multiplication the two original matrices are not multiplied, but the matrix that would go in the denominator and that now multiplies is the inverse matrix of the original matrix.

Matrix division formula

Matrix Division Formula
Matrix division formula

The inverse matrix is ​​made over the denominator matrix.

Matrix division process

The order to divide two matrices is as follows:

  1. Determine which matrix goes in the numerator and which matrix goes in the denominator. Remember that the denominator matrix has to be invertible. Otherwise, the division cannot be made.
  2. Make the inverse of the matrix that goes in the denominator.
  3. Multiply the numerator matrix by the inverse matrix.
  4. Smile because we have done well!

Theoretical example

Given any two matrices,

Arrays
Arrays

Putting the above matrices in the following form:

Matrix Division 2
Matrix division

In this case we would be dividing matrix A by matrix C.

So if we want to use matrix C as a dividing matrix, what should we check first? Exactly, if this matrix is ​​invertible or not.

Conditions for a matrix to be inverse

The conditions are:

  1. The matrix has to be a square matrix.
  2. The determinant of the matrix has to be different from zero (0).

Next, we evaluate if we can continue with the division of matrices or not:

  • If matrix C can be an inverse matrix, we continue with division.
  • If matrix C cannot be an inverse matrix because it does not meet the conditions, we cannot continue the division with this matrix as a denominator or divider matrix.

Practical example

Given the following matrices, divide matrix X by matrix B :

Matrices 1
Arrays

We first determine which matrix goes in the numerator and which matrix goes in the denominator. This condition is given by the statement, in this example, matrix X would be the dividend matrix or numerator matrix and matrix B would be the divisor matrix or denominator matrix.

  • X matrix → Dividend matrix or denominator matrix.
  • Matrix B → Divider matrix or denominator matrix.

Second, we check that we can do the inverse of the matrix that goes in the denominator, in this case, matrix B.

Matrix B is a square matrix and the determinant is different from zero (0), therefore the inverse matrix of matrix B exists and is denoted as B -1 .

Inverse Matrix Of Matrix B
Inverse matrix of matrix B

Third, we multiply matrix X by matrix B -1 .

Matrix Division
Matrix division

Fourth, we smile because we have done the matrix division right!