Linear Systems in Matlab
Contents
Solving Linear Systems of Equations in MATLAB
This section discusses how to solve a set of linear equations in MATLAB. See the discussion of linear algebra for help on writing a linear system of equations in matrixvector format. There is also help on creating matrices and vectors in MATLAB.
The simplest way of solving a system of equations in MATLAB is by using the \ operator. Given a matrix A and a vector b, we may solve the system using the following MATLAB commands:
x = A\b; % Solve the linear system Ax=b for x.
Example
Consider the following set of equations:These can be easily solved by hand to obtain . These equations and their solution (intersection) are plotted in the figure to the right.
To solve the system of equations using MATLAB, first rewrite these in a matrixvector form as
Once in matrixvector form, the solution is obtained in MATLAB by using the following commands (see here for help on creating matrices and vectors):
A = [ 5 6; 3 1 ]; % define the matrix
b = [ 0; 5 ]; % define the vector
solution = A\b; % solve the system of equations.
In this example, solution is a column vector whose elements are x and y:
which is consistent with the answer we obtained by hand above.
Matrix Inverse
Note that we can also form the inverse of a matrix,
This can be done in MATLAB as illustrated by the following:
A = [ 5 6; 3 1 ];
b = [ 0; 5 ];
Ainv = A^1; % calculate the inverse of A
solution = Ainv*b; % calculate the solution
We could also calculate A^{1} by
Ainv = inv(A); % entirely equivalent to A^1.
Sparse Systems

Linear Systems using the Symbolic Toolbox
Occasionally we may want to find the symbolic (general) solution to a system of equations rather than a specific numerical solution. The symbolic toolbox provides a way to do this.
