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 \(\text{rank}(A) = \text{rank}([A \mid B])\). System is consistent: This means there exists a solution \(X\) such that \(AX = B\). Augmented Matrix Representation: The system \(AX = B\) can be written as an augmented matrix \([A \mid B]\). Implication for ranks: Since \(AX = B\) has a solution, the vector \(B\) lies in the column space of \(A\). Therefore, the addition of...