Consistency in Equation Concept

TL;DR

Consistency of a non-homogeneous linear equation is determined by comparing the ranks of the coefficient matrix and the augmented matrix.

Transcript

hi everyone today we are going to discuss concept of consistency in non-homogeneous linear equation condition for consistency consider consider m equation m equation in n unknowns in n unknowns therefore therefore the system of the system of nonhomogeneous linear equation linear equation given as therefore the system of non-homogeneous linear equat... Read More

Key Insights

• 💁 Non-homogeneous linear equations can be represented in matrix form using a coefficient matrix, unknown matrix, and constant matrix.
• 😜 The condition for consistency in non-homogeneous linear equations is determined by comparing the ranks of the coefficient matrix and the augmented matrix.
• 😜 Consistent systems have either a unique solution or infinitely many solutions, depending on the rank relationship.
• ❓ Inconsistent systems have no solution.

Questions & Answers

Q: What is the definition of a consistent non-homogeneous linear equation?

A non-homogeneous linear equation is consistent if it has one or more solutions. It means the ranks of the coefficient matrix and the augmented matrix are the same.

Q: How do you determine the consistency of a non-homogeneous linear equation using the rank of matrices?

The system is consistent if the rank of the coefficient matrix is equal to the rank of the augmented matrix. If the ranks are different, the system is inconsistent and has no solution.

Q: What are the two cases for a consistent non-homogeneous linear equation?

In the first case, if the ranks of the matrices are equal to the number of unknowns, the system has a unique solution. In the second case, if the rank of the coefficient matrix is less than the number of unknowns, the system has infinitely many solutions.

Q: How can you express the solutions of a non-homogeneous linear equation with infinitely many solutions?

When the system has infinitely many solutions, the values can be expressed in terms of a parameter. By assigning an arbitrary value to the parameter, the remaining variables can be determined.

Summary & Key Takeaways

• The content discusses the concept of consistency in non-homogeneous linear equations, explaining the conditions and criteria for a system to be consistent.

• It introduces the coefficient matrix, unknown matrix, and constant matrix, and how they are used to represent the system in matrix form.

• The content provides a chart to determine the consistency and solution types based on the ranks of the matrices involved.