Proof of Consistency Condition for Linear System

Proof of Consistency Condition for Linear System

Proof of Consistency Condition for Linear System

To prove that a necessary and sufficient condition for the system of linear equations \(AX = B\) to be consistent is that the matrices \(A\) and \([A \mid B]\) have the same rank, we will use the rank criterion for the consistency of linear systems.

Necessary Condition

First, we show that if the system \(AX = B\) is consistent, then \(\text{rank}(A) = \text{rank}([A \mid B])\).

  1. System is consistent: This means there exists a solution \(X\) such that \(AX = B\).
  2. Augmented Matrix Representation: The system \(AX = B\) can be written as an augmented matrix \([A \mid B]\).
  3. Implication for ranks:
    • Since \(AX = B\) has a solution, the vector \(B\) lies in the column space of \(A\).
    • Therefore, the addition of \(B\) as an extra column to \(A\) does not increase the rank. This implies that \(\text{rank}(A) = \text{rank}([A \mid B])\).

Sufficient Condition

Now, we show that if \(\text{rank}(A) = \text{rank}([A \mid B])\), then the system \(AX = B\) is consistent.

  1. Assume \(\text{rank}(A) = \text{rank}([A \mid B])\): Let \(r = \text{rank}(A)\).
  2. Rank Definition:
    • The rank of a matrix is the maximum number of linearly independent rows (or columns) in the matrix.
    • If \(\text{rank}(A) = r\), then there are \(r\) linearly independent rows in \(A\).
    • Since \(\text{rank}([A \mid B]) = r\), the addition of \(B\) as an extra column does not increase the rank.
  3. Consistency:
    • The fact that the rank does not increase implies that \(B\) can be written as a linear combination of the columns of \(A\).
    • Therefore, there exists a solution \(X\) such that \(AX = B\), implying the system is consistent.

Conclusion

We have shown both:

  • If the system \(AX = B\) is consistent, then \(\text{rank}(A) = \text{rank}([A \mid B])\).
  • If \(\text{rank}(A) = \text{rank}([A \mid B])\), then the system \(AX = B\) is consistent.

Thus, the necessary and sufficient condition for the system of equations \(AX = B\) to be consistent is that the matrices \(A\) and \([A \mid B]\) have the same rank.

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