Solving Systems using Matrices:
A matrix is an array of numbers arranged in rows and columns.
ex.
This is a 2 by 3 matrix, meaning there are 2 rows and 3 columns
Generally, matrices of x rows and y columns are called dimension x × y or order x × y.
With every system of equation, there is a matrix of the coefficients and constant term that is associated with the system.
ex.
The system of equation:
3x + 5y = -4
x - 3y = 5
has a matrix that looks like;
This matrix is commonly called an augmented matrix, the dotted line separates the coefficients and the constant terms.
For any augmented matrices of linear system of equations, the following operations will produce an equivalent matrix:
1. Interchanging any two rows of the matrix.
2. Multiplying a row with any nonzero real number.
3. Any row of matrix can be changed by adding a nonzero multiple of
another row to that row.
With the operations above, reducing the matrix into a reduced echelon form will solve the system.
ex.
from the example;
2x + 3y + z = 1
5x + 2y - 3z = 8
x - 4y - z = 18
the matrix looks like:
To solve the matrix;
The solution implies that x = 5, y = -4 and z = 3, which is similar to the previous example.
The solution set is {(5,-4,3)}.
