Solving 3x3 Linear Systems with Matrices: Inconsistent Systems and No Solutions

In the realm of mathematics, linear systems of equations are a fundamental concept, finding applications in diverse fields like engineering, economics, and computer science. A linear system is a set of equations where each equation represents a straight line or plane in a multidimensional space. Solving such systems involves finding the values of the variables that satisfy all the equations simultaneously.

One powerful method for solving linear systems is using matrices. Matrices provide a compact and efficient way to represent and manipulate linear equations. This blog post delves into the process of solving 3x3 linear systems with matrices, specifically focusing on inconsistent systems that have no solutions.

Understanding Inconsistent Systems

An inconsistent system of equations is a system where there is no set of values for the variables that satisfies all equations simultaneously. Geometrically, this means that the lines or planes represented by the equations do not intersect at a common point. For a 3x3 system, this translates to three planes that do not intersect at a single point.

Solving 3x3 Linear Systems with Matrices

Let's consider a general 3x3 linear system:

a₁₁x + a₁₂y + a₁₃z = b₁

a₂₁x + a₂₂y + a₂₃z = b₂

a₃₁x + a₃₂y + a₃₃z = b₃

We can represent this system using an augmented matrix:

[ A | B ] =
[ a₁₁ a₁₂ a₁₃ | b₁ ]
[ a₂₁ a₂₂ a₂₃ | b₂ ]
[ a₃₁ a₃₂ a₃₃ | b₃ ]

Where A is the coefficient matrix and B is the constant vector.

Identifying Inconsistent Systems

To determine if a 3x3 system is inconsistent, we can use Gaussian elimination, a method for transforming the augmented matrix into row echelon form. In row echelon form, the matrix has the following properties:

• The first non-zero element in each row (leading entry) is 1.
• The leading entry in each row is to the right of the leading entry in the row above it.
• Rows with all zero elements are at the bottom of the matrix.

If, during Gaussian elimination, we encounter a row with all zeros except for the last element (which is non-zero), then the system is inconsistent. This indicates a contradiction, meaning there is no solution that satisfies all equations.

Example: Inconsistent System

Let's consider the following system:

x + 2y - z = 1

2x + 4y - 2z = 3

x - y + z = 2

The augmented matrix for this system is:

[ A | B ] =
[ 1 2 -1 | 1 ]
[ 2 4 -2 | 3 ]
[ 1 -1 1 | 2 ]

Performing Gaussian elimination, we obtain:

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

Notice that the second row has all zeros except for the last element, which is 1. This signifies a contradiction, indicating that the system is inconsistent and has no solutions.

Conclusion

Solving 3x3 linear systems with matrices is a powerful and efficient method. By utilizing Gaussian elimination, we can determine if a system is consistent or inconsistent. Inconsistent systems, which have no solutions, arise when the equations represent lines or planes that do not intersect at a common point. Understanding inconsistent systems is crucial for solving various mathematical problems and real-world applications.