22.7.1. Theorem: Rouché–Capelli Theorem

Try these first

1. 1.

   Define the Rouché–Capelli Theorem.

   Hint

   Focus on the relationship between ranks.

2. 2.

   What does it mean if rank(A) ≠ rank([A∨B])?

   Hint

   Think about non-matching conditions.

3. 3.

   What does the Rouché–Capelli Theorem determine?

   • The number of solutions
   • The rank of a matrix
   • The determinant of a matrix

   Hint

   Focus on what the theorem facilitates.

4. 4.

   True or False: If rank(A) = 3 and rank([A∨B]) = 3, the system has no solutions.

   • True
   • False

   Hint

   Think about the implications of rank equality.

5. 5.

   Given the system: 3x + 2y = 5, 6x + 4y = 10, analyze whether the system is consistent or inconsistent and find the nature of the solutions.

   Hint

   Look for rank similarity in matrix forms.

6. 6.

   A matrix A has rank 4 and is used in a system of 6 variables. Determine the type of solution the system provides.

   Hint

   Recall how ranks impact solution types.