Problem 3 Based on Homogenous Equations

TL;DR

Learn how to solve a homogeneous system of equations using matrix form and row transformations.

Transcript

hi everyone today we are going to discuss problem number three based on homogeneous equation that is in homogeneous system of equation rhs is zero so let's see so problem is here solve the following system of equations we have to given x plus y minus z plus w equal to zero second equation x minus y plus two z minus w equal to zero third equation th... Read More

Key Insights

• 🫱 The given problem involves a homogeneous system of equations with a zero right-hand side.
• 🚱 The number of equations and unknowns are different, indicating a non-trivial solution.
• 💁 The system is converted into matrix form, and row transformations are used to reduce it to echelon form.
• 😜 The rank of the matrix determines the number of solutions.
• 😜 In this case, the rank is less than the number of unknowns, leading to infinitely many solutions.
• 🚱 Two parameters are introduced to represent the non-trivial solution.

Questions & Answers

Q: How do you determine if a system of equations is homogeneous?

A system of equations is homogeneous if the right-hand side of each equation is zero.

Q: What is the purpose of writing the system of equations in matrix form?

Writing the system in matrix form allows for easier manipulation and solving using matrix operations.

Q: What is the coefficient matrix in the matrix form of the system?

The coefficient matrix contains the coefficients of the variables in each equation, arranged in columns.

Q: How is the augmented matrix of the system obtained?

The augmented matrix is obtained by combining the coefficient matrix with a null matrix on the right-hand side.

Q: How do you perform row transformations to reduce the matrix to echelon form?

Row transformations involve operations on rows to eliminate coefficients below the leading element and make the leading element one.

Q: How do you determine the rank of the matrix?

The rank of the matrix is the number of non-zero rows in the echelon form.

Q: What is the significance of the rank in determining the number of solutions?

If the rank is less than the number of unknowns, the system has infinitely many solutions.

Q: How do you find the non-trivial solution when the rank is less than the number of unknowns?

You assign parameters to the remaining variables and express the solution in terms of those parameters.

Summary & Key Takeaways

• The given problem involves solving a system of equations with a zero right-hand side, making it a homogeneous system.

• The number of equations is 3, and the number of unknowns is 4.

• The system is converted into matrix form and an augmented matrix is created.