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In the matrix equation Px = q, which of the following is a necessary condition for existence of at least one solution for unknown vector x?

In the matrix equation Px = q, which of the following is a necessary condition for existence of at least one solution for unknown vector x? Correct Answer Augmented matrix [Pq] must have the same rank as matrix P.

Concept:

  • The equation Px = q will have at least one solution only if it is consistent.
  • The given system will be consistent only if Rank = Rank .​

​Note:

So, a necessary condition for the existence of at least one solution for unknown vector x is that the augmented matrix must have the same rank as matrix P.

Additional Information

A consistent solution means either the system has a unique solution or infinitely many solutions. So, 

  1. the matrix Px = q will have a unique solution if Rank = Rank = m, where 'm' is of order of matrix P, and
  2. the matrix Px = q will have infinitely many solutions if Rank = rank < m. 

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