How to Solve Homogenous Equations

TL;DR

Learn how to solve homogeneous system of linear equations by understanding the conditions for consistency and determining the types of solutions.

Transcript

hi everyone today we are going to discuss system of homogeneous linear equations means we have to solve here how to solve homogeneous linear equations now let me start how to solve homogeneous system of linear equations so especially you know that when your system is in matrix form is in matrix form that is x is equal to b this is a matrix form of ... Read More

Key Insights

• 🚱 Homogeneous systems of linear equations can be represented in matrix form as Ax=B, where B determines if the system is homogeneous or non-homogeneous.
• 💁 To solve a homogeneous system, it needs to be converted into matrix form and then into augmented matrix form.
• 😜 The rank of the coefficient matrix and the augmented matrix determines the consistency of the system.
• 😜 A homogeneous system can have a unique solution or infinitely many solutions, depending on the rank and determinant of the coefficient matrix.

Questions & Answers

Q: What is the difference between a homogeneous and a non-homogeneous system of linear equations?

In a homogeneous system, the matrix B is equal to zero, while in a non-homogeneous system, the matrix B is not equal to zero. This distinction determines the consistency of the system.

Q: How can we determine if a homogeneous system is consistent?

By comparing the rank of the coefficient matrix with the rank of the augmented matrix. If they are equal, the system is consistent. If the rank of the coefficient matrix is less than the number of unknowns, the system has infinite solutions.

Q: Can a homogeneous system have a unique solution?

Yes, if the rank of the coefficient matrix is equal to the number of unknowns, the system has a unique solution, also known as a trivial solution.

Q: How can we determine the types of solutions for a homogeneous system?

For a homogeneous system with m=n, we can check the determinant of the coefficient matrix. If it is not equal to zero, the system has a trivial solution. If it is equal to zero, the system has infinite solutions.

Summary & Key Takeaways

• Homogeneous system of linear equations can be represented as a matrix. It can be classified as either homogeneous or non-homogeneous based on the value of the matrix B.

• The process of solving homogeneous equations involves converting the system into matrix form and then into augmented matrix form.

• There are two types of solutions for homogeneous systems: trivial solution (unique solution) and non-trivial solution (infinite solutions).