A system of linear non-homogeneous equations is consistent if and only if the rank of the coefficient matrix is equal to the rank of __.
Understand the Problem
The question is asking for a condition that makes a system of linear non-homogeneous equations consistent. Specifically, it is asking for the relationship between the rank of the coefficient matrix and another rank in order for the system to be consistent.
Answer
A system of linear non-homogeneous equations is consistent if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix: $$ \text{rank}(A) = \text{rank}([A|b]). $$
Answer for screen readers
A system of linear non-homogeneous equations is consistent if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix: $$ \text{rank}(A) = \text{rank}([A|b]) $$
Steps to Solve
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Understanding the system of equations To determine the consistency of a system of linear non-homogeneous equations, we look at two ranks: the rank of the coefficient matrix ( A ) and the rank of the augmented matrix ( [A|b] ).
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Apply the Rank Condition A system of linear equations is consistent if the rank of the coefficient matrix is equal to the rank of the augmented matrix. This can be expressed mathematically as: $$ \text{rank}(A) = \text{rank}([A|b]) $$
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Stating the Condition Thus, the answer to the question is that a system of linear non-homogeneous equations is consistent if and only if: $$ \text{rank}(A) = \text{rank}([A|b]) $$
A system of linear non-homogeneous equations is consistent if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix: $$ \text{rank}(A) = \text{rank}([A|b]) $$
More Information
This rank condition is fundamental in linear algebra. A non-homogeneous system might have no solutions, one solution, or infinitely many solutions, with consistency indicating the presence of at least one solution.
Tips
- Misunderstanding the roles of the ranks: It's easy to confuse the purpose of the ranks. Remember, rank of the coefficient matrix indicates the maximum number of linearly independent rows (or columns), while the rank of the augmented matrix adds the information from the constant terms.
- Assuming all systems are consistent: Just because a system appears solvable doesn’t mean it's consistent. Always check the rank condition.
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