The condition for consistency of simultaneous equation AX = B is?

Question image

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

  1. 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 $[A|B]$.

  2. 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.
  1. 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.

  2. 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.
Thank you for voting!
Use Quizgecko on...
Browser
Browser