A Linear System can be represented using a Matrix. This method is especially useful in computer programs, as matrices can be solved incredibly fast by a computer.
Via Coefficient & Augmented Matrices
A linear system can be represented with a coefficient matrix to show the coefficients of the various variables, augmented with a constant column matrix to represent the constants
For example, the linear system:
Can be written with a coefficient matrix as:
As the system has 2 equations and 3 variables, resulting in a matrix.
We can augment the existing coefficient matrix to show the constant terms. Again, with the example above:
would be the resulting augmented matrix. (The are not part of the matrix, just useful to identify which coefficient belongs to which variable.)
The theorem of consistency applies here.
Solving
This form can be solved using Gauss-Jordan Elimination, to obtain a Reduced Row Echelon Form which can be easily solved.
Alternatively, the Gaussian Elimination can be used to obtain a Row Echelon Form, which can then be solved via Back Substitution.
Via Matrix Multiplication
A linear system can also be represented in the form:
Where and are both column matrices. For example if ,
Then:
which translates to:
Solving
This form of linear system can be solved by obtaining the Inverse Matrix, , and using Matrix Multiplication:
which can be trivially solved.
Info
Once we obtain the inverse, , any version of works, as the final solution is given by
#todo examples