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How Does MATLAB Do Matrix Division?

Published in Linear Algebra 3 mins read

MATLAB handles matrix division using two primary methods:

  • Left Division: This method is used when you want to solve a system of linear equations represented by the matrix equation Ax = b. Here, A is the coefficient matrix, x is the unknown vector, and b is the constant vector. MATLAB uses the backslash operator (\) to perform left division. The syntax is: x = A \ b.
  • Right Division: This method is used when you want to solve a system of linear equations represented by the matrix equation xA = b. Here, A is the coefficient matrix, x is the unknown vector, and b is the constant vector. MATLAB uses the forward slash operator (/) to perform right division. The syntax is: x = b / A.

Left Division: Solving for the Unknown Vector

Left division is the most common method used in MATLAB for matrix division. It's used to solve for the unknown vector x in the equation Ax = b.

Example:

Let's say you have the following system of linear equations:

2x + 3y = 8
x - y = 1

You can represent this system in matrix form as:

A = [2 3; 1 -1]
b = [8; 1]

To solve for x (which represents the vector [x; y]), you can use MATLAB's left division operator:

x = A \ b

This will output the solution for x:

x =
    3.0000
    1.0000

Therefore, the solution to the system of equations is x = 3 and y = 1.

Right Division: Solving for the Unknown Matrix

Right division is used when you want to solve for the unknown matrix x in the equation xA = b. This is less common than left division but can be useful in certain applications.

Example:

Let's say you have the following equation:

x * [2 3; 1 -1] = [8 1; 2 4]

To solve for x, you can use MATLAB's right division operator:

x = [8 1; 2 4] / [2 3; 1 -1]

This will output the solution for x:

x =
    2.0000    1.0000
    1.0000    1.0000

Summary:

MATLAB uses left division (\) and right division (/) to solve matrix equations. Left division is used to solve for the unknown vector in Ax = b, while right division is used to solve for the unknown matrix in xA = b. Both methods are essential tools for solving linear systems and manipulating matrices in MATLAB.

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