Problem 4 Based on Consistency in Equation

TL;DR

This content discusses the concept of consistency in equations, demonstrating how to determine if a system of equations is consistent and finding its solution if it is consistent.

Transcript

hi everyone today we are going to discuss problem number four based on consistency in equation so when we think about consistency if system of equation is a consistent then it has a solution and if it has a solution there has it may be unique solution or it may be infinitely many solution and if consistency is not is there means system is inconsist... Read More

Key Insights

• ❓ Consistency in equations refers to whether a system of equations has a solution.
• ❓ The coefficient matrix represents the coefficients of the equations, and the constant matrix represents the constants.
• 💁 By reducing the augmented matrix to equilibrium form, the consistency of the system can be determined.
• 😜 If the ranks of the coefficient matrix and the augmented matrix are equal, the system is consistent.
• 😜 If the ranks are not equal, the system is inconsistent and has no solution.
• 🥺 The process involves performing row transformations to make the leading elements 1 and the elements below the leading elements 0.
• 😜 The rank of a matrix is determined by the number of non-zero rows in the matrix.

Questions & Answers

Q: What is the minimum condition for a system of equations to be consistent?

The minimum condition for consistency is that the rank of the coefficient matrix (matrix A) must be equal to the rank of the augmented matrix (matrix A|B).

Q: How can the system of equations be written in matrix form?

The system of equations can be written in matrix form as X = B, where X is the unknown matrix (containing variables x, y, z) and B is the constant matrix.

Q: What is the purpose of reducing the augmented matrix to equilibrium form?

By performing row transformations to make the leading elements 1 and the elements below the leading elements 0, the equilibrium form of the augmented matrix helps determine the consistency of the system of equations.

Q: How is the consistency of a system of equations determined?

The consistency of a system of equations is determined by comparing the ranks of the coefficient matrix and the augmented matrix. If the ranks are not equal, the system is inconsistent.

Summary & Key Takeaways

• The content explains the concept of consistency in equations and the conditions for a system of equations to be consistent.

• It demonstrates how to write a system of equations in matrix form and construct the coefficient matrix and constant matrix.

• The process of reducing the augmented matrix to equilibrium form is explained, followed by a determination of whether the system of equations is consistent or inconsistent.