22.7. Consistency of a Linear System: Rank-Based Approach

Try these first

1. 1.

   What is the definition of a matrix rank?

   Hint

   Consider the relationship of rows and columns.

2. 2.

   In the context of a linear system, what does an inconsistent system mean?

   Hint

   Think about the relationships between the equations.

3. 3.

   What does it mean for a system to be consistent?

   • It has no solutions
   • It has at least one solution
   • It has infinitely many solutions

   Hint

   Recall the definitions of consistent and inconsistent systems.

4. 4.

   True or False: If rank(A) = rank([A∨B]), the system is always consistent.

   • True
   • False

   Hint

   Think about what this theorem states.

5. 5.

   Consider the system: x + y + z = 1; x + y + z = 2. Analyze the rank and determine if it is consistent. Explain why or why not.

   Hint

   Consider the meaning of parallel lines in a graph.

6. 6.

   Given the equations: 3x + 4y = 5 and 6x + 8y = 10, what can you deduce about the rank and the solutions of this system?

   Hint

   Reflect on how multiplicative relations between equations affect rank.