The condition for consistency of simultaneous equation AX = B is?
Understand the Problem
The question is asking about the condition for the consistency of simultaneous equations of the form AX = B, specifically which rank condition must be satisfied for the solution to exist.
Answer
The condition for consistency is $ \text{Rank A} = \text{Rank B} $.
Answer for screen readers
The condition for consistency of the simultaneous equation $AX = B$ is: $$ \text{Rank A} = \text{Rank B} $$
Steps to Solve
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Understanding Rank Condition The rank condition for the consistency of the equation $AX = B$ states that the rank of the coefficient matrix $A$ must be equal to the rank of the augmented matrix $[A|B]$.
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Identify the Ranks Let:
- $r_A$ be the rank of matrix $A$.
- $r_B$ be the rank of the augmented matrix $[A|B]$.
- $r_C$ be the rank of matrix $C$, which is generally the ranks of the columns in the equation.
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Applying the Consistency Condition For the system to have a solution (be consistent), the condition is: $$ r_A = r_B $$ This means the rank of the coefficient matrix must equal the rank of the augmented matrix.
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Choosing the Correct Option Since the problem asks for the correct rank condition, we compare:
- If $r_A$ (Rank of $A$) equals $r_B$ (Rank of $B$), then it ensures $AX = B$ is consistent.
The condition for consistency of the simultaneous equation $AX = B$ is: $$ \text{Rank A} = \text{Rank B} $$
More Information
In linear algebra, the rank of a matrix reflects the dimension of the vector space generated by its rows or columns. For a system of equations, a higher rank indicates more independent equations.
Tips
- Assuming that $r_A = r_C$ is a condition for consistency without realizing that it is $r_A = r_B$ that matters.
- Not distinguishing between the different ranks can lead to confusion in the application of the consistency condition.
