Final Form of a 3x3 Augmented Matrix: Mathematical Solution

_{October 3, 2023 by JoyAnswer.org, Category : Mathematics}

What is the final form of a 3x3 augmented matrix? Explore the mathematical process to determine the final form of a 3x3 augmented matrix when using Gaussian elimination for solving linear systems.

What is the final form of a 3x3 augmented matrix?

The final form of a 3x3 augmented matrix depends on the specific system of linear equations it represents and the operations applied to it during the process of solving the system using Gaussian elimination or row reduction. The goal is to transform the augmented matrix into one of the following three forms:

Row-Echelon Form (REF):
- In row-echelon form, the matrix has the following properties:
  - The first nonzero entry in each row (called the leading coefficient) is 1.
  - The leading coefficient in each row is to the right of the leading coefficient in the row just above it.
  - Rows consisting of all zeros, if any, are at the bottom of the matrix.
The last column in the row-echelon form contains the solutions to the system of equations.
Example of a 3x3 augmented matrix in row-echelon form:
```
1  2  0  |  5
0  1  3  | -2
0  0  1  |  3
```
Reduced Row-Echelon Form (RREF):
- In reduced row-echelon form, the matrix has the properties of row-echelon form, and:
  - All other entries in the column containing a leading coefficient are zeros.
  - The leading coefficient in each row is the only nonzero entry in its column.
Example of a 3x3 augmented matrix in reduced row-echelon form:
```
1  0  0  |  2
0  1  0  | -3
0  0  1  |  1
```
Inconsistent Form:
- If the system of equations has no solution, the augmented matrix may have a row that represents a contradictory equation, such as 0 0 0 | 3, indicating that there is no solution.
Example of a 3x3 augmented matrix for an inconsistent system:
```
1  2  3  |  4
0  1  2  |  3
0  0  0  |  1
```
Infinitely Many Solutions Form:
- In some cases, the system of equations may have infinitely many solutions. The augmented matrix will have at least one row representing a dependent equation, such as 0 0 0 | 0, indicating that the equation is dependent on others, leading to infinitely many solutions.
Example of a 3x3 augmented matrix for a system with infinitely many solutions:
```
1  2  3  |  4
0  1  2  |  3
0  0  0  |  0
```

The final form of the 3x3 augmented matrix will depend on the specific problem you are trying to solve and the steps taken during the Gaussian elimination or row reduction process. It may be in row-echelon form, reduced row-echelon form, or one of the forms indicating no solution or infinitely many solutions.

The final form of a 3x3 augmented matrix is a matrix in which the leading coefficient of each row is 1 and the other entries in the row are 0. This is achieved using a process called Gaussian elimination, which involves performing elementary row operations until the matrix is in reduced row echelon form.

Here is an example of a 3x3 augmented matrix in reduced row echelon form:

[ 1 0 0 | 3 ]
[ 0 1 0 | 2 ]
[ 0 0 1 | 1 ]

This matrix indicates that the system of equations has a unique solution, which is x = 3, y = 2, and z = 1.

Another example of a 3x3 augmented matrix in reduced row echelon form is:

[ 1 0 0 | 3 ]
[ 0 1 0 | 2 ]
[ 0 0 0 | 0 ]

This matrix indicates that the system of equations has infinitely many solutions.

If the augmented matrix is not in reduced row echelon form, then the system of equations either has no solution or infinitely many solutions.

Here are some tips for finding the final form of a 3x3 augmented matrix:

Use Gaussian elimination to eliminate the leading coefficients of the second and third rows.
If the leading coefficient of a row is 0, then switch the row with another row that has a non-zero leading coefficient.
Use back substitution to solve for the variables.

If you are having trouble finding the final form of a 3x3 augmented matrix, you can use a matrix calculator or ask for help from a tutor or teacher.