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Consider a non-homogeneous system of linear equations representing mathematically an over-determined system. Such a system will be
- a)Consistent having a unique solution
- b)Consistent having many solutions
- c)Inconsistent having unique solution
- d)Inconsistent having no solution
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Consider a non-homogeneous system of linear equations representing ma...
Determined system In this system, the number of equations is more than number of unknown variables, i.e., (m>n)
So, in determined system it is necessary that the system should be consistent and having unique solution.
Most Upvoted Answer
Consider a non-homogeneous system of linear equations representing ma...
Introduction:
A system of linear equations is considered over-determined when there are more equations than unknowns. In this case, the system is inconsistent, meaning that there is no solution that satisfies all the equations. However, if the system is non-homogeneous, it is still possible for it to have a unique solution or infinitely many solutions.
Explanation:
Let's consider a non-homogeneous system of linear equations with more equations than unknowns. Such a system can be represented as follows:
Equation 1: a₁₁x₁ + a₁₂x₂ + ... + a₁ₙxₙ = b₁
Equation 2: a₂₁x₁ + a₂₂x₂ + ... + a₂ₙxₙ = b₂
...
Equation m: aₘ₁x₁ + aₘ₂x₂ + ... + aₘₙxₙ = bₘ
Here, m > n, where m represents the number of equations and n represents the number of unknowns.
Consistency and Uniqueness of Solution:
In an over-determined system, it is not possible for all the equations to be satisfied simultaneously. However, the non-homogeneous nature of the system allows for the possibility of a unique solution.
If the coefficient matrix of the system has full rank (i.e., rank(A) = n), then the system is consistent and has a unique solution. The full rank condition ensures that there are no redundant equations.
In this case, even though there are more equations than unknowns, the system can still be consistent because the equations are not contradictory.
Inconsistency of Solution:
If the coefficient matrix of the system does not have full rank (i.e., rank(A) < n),="" then="" the="" system="" is="" inconsistent="" and="" has="" no="" solution.="" this="" means="" that="" the="" equations="" are="" contradictory="" and="" cannot="" be="" satisfied="" />
The inconsistency arises due to the overdetermined nature of the system, where there are more equations than unknowns. This leads to an insufficient number of variables to satisfy all the equations.
Conclusion:
In summary, a non-homogeneous system of linear equations representing an over-determined system can be consistent with a unique solution if the coefficient matrix has full rank. However, if the coefficient matrix does not have full rank, the system is inconsistent and has no solution. Therefore, the correct answer is option 'A' - Consistent having a unique solution.
A system of linear equations is considered over-determined when there are more equations than unknowns. In this case, the system is inconsistent, meaning that there is no solution that satisfies all the equations. However, if the system is non-homogeneous, it is still possible for it to have a unique solution or infinitely many solutions.
Explanation:
Let's consider a non-homogeneous system of linear equations with more equations than unknowns. Such a system can be represented as follows:
Equation 1: a₁₁x₁ + a₁₂x₂ + ... + a₁ₙxₙ = b₁
Equation 2: a₂₁x₁ + a₂₂x₂ + ... + a₂ₙxₙ = b₂
...
Equation m: aₘ₁x₁ + aₘ₂x₂ + ... + aₘₙxₙ = bₘ
Here, m > n, where m represents the number of equations and n represents the number of unknowns.
Consistency and Uniqueness of Solution:
In an over-determined system, it is not possible for all the equations to be satisfied simultaneously. However, the non-homogeneous nature of the system allows for the possibility of a unique solution.
If the coefficient matrix of the system has full rank (i.e., rank(A) = n), then the system is consistent and has a unique solution. The full rank condition ensures that there are no redundant equations.
In this case, even though there are more equations than unknowns, the system can still be consistent because the equations are not contradictory.
Inconsistency of Solution:
If the coefficient matrix of the system does not have full rank (i.e., rank(A) < n),="" then="" the="" system="" is="" inconsistent="" and="" has="" no="" solution.="" this="" means="" that="" the="" equations="" are="" contradictory="" and="" cannot="" be="" satisfied="" />
The inconsistency arises due to the overdetermined nature of the system, where there are more equations than unknowns. This leads to an insufficient number of variables to satisfy all the equations.
Conclusion:
In summary, a non-homogeneous system of linear equations representing an over-determined system can be consistent with a unique solution if the coefficient matrix has full rank. However, if the coefficient matrix does not have full rank, the system is inconsistent and has no solution. Therefore, the correct answer is option 'A' - Consistent having a unique solution.
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Consider a non-homogeneous system of linear equations representing mathematically an over-determined system. Such a system will bea)Consistent having a unique solutionb)Consistent having many solutionsc)Inconsistent having unique solutiond)Inconsistent having no solutionCorrect answer is option 'A'. Can you explain this answer?
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