Linear Equation System in the Form of Matrix
An arbitrary system
consisting of ni linear equations with n unknown numbers will be written
as,
The system of linear equations can be written in the form of a matrix,
This line is called an enlarged matrix or an augmented matrix
Example 1
can be represented by:
The basic method for solving a system of linear equations is to replace the system provided with a new system that has the same set of solutions with easier solutions. This new system is generally obtained in a stage by applying operations to eliminate systematically unknown numbers. These operations are called Elementary Line Operations (OBE), which will be discussed below,
SETTLEMENT OF THE LINEAR EQUATION SYSTEM WITH ELEMENTER LINE OPERATIONS
The basic method used to solve a system of known linear equations is substitution techniques, elimination, and so on. For large linear equation systems, elimination techniques will be more effective than other techniques.
The idea of an elimination technique arises by changing a system of linear equations into a system of other equal linear equations (in the sense that the solution does not change), but which has a form that is easily solved. Systematic steps in elimination are:
- Multiply equations with nonzero constants
- Exchange two lines of equations
- Add multiples of one equation to another equation.
Example 1
In the system of equations:
Add the first line multiplied by -2 to the second row to get,
Add the first line multiplied by -3 to the third row to get,
Multiply the second equation with 1/2 to get,
Add the second line multiplied by -3 to the third row to get,
Multiply the third equation with -2 to get,
Add the second line multiplied by -1 in the first row to get,
Add the third line multiplied by -11/2 to the first row and the third row multiplied by 7/2 to the second row to get,
Because the rows in the complete matrix correspond to the equations in the system associated with the line, the three operations above correspond to the following operations in the complete matrix, namely:
- Multiply a line with a non-zero constant.
- Exchange the two lines.
- Add multiplication from one line to another.
Example 3.
In the system of equations:
Step 1,
The enlarged matrix for this system is,
Step 2,
Add the first line multiplied by -2 to the second row to get,
Step 3,
Add the first line multiplied by -3 to the third row to get,
Step 4,
Multiply the second row with 1/2 to get,
Step 5,
Add the second line multiplied by -3 to the third row to get,
Step 6,
Multiply the third line with -2 to get,
Step 7,
Add the second line multiplied by -1 in the first row to get,
Step 8,
Add the third line multiplied by -11/2 to the first row and the third row multiplied by 7/2 to the second row to get,
So the solution, we get: x = 1, y = 2, z = 3.
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