25.3. Conditions for Existence of a Solution

Try these first

1. 1.

   What is the condition for a system of linear equations to be consistent?

   Hint

   Think about the relationship between the ranks.

2. 2.

   If Rank(A) is 2 and Rank([A|b]) is 2, what can we say about the system?

   Hint

   Check how ranks relate to solutions.

3. 3.

   What condition must hold for a linear system to have at least one solution?

   • Rank(A) ≠ Rank([A|b])
   • Rank(A) = Rank([A|b])
   • Rank(A) > Rank([A|b])

   Hint

   Think about the equality between ranks.

4. 4.

   True or False: A system of linear equations can still be consistent if Rank(A) is less than Rank([A|b]).

   • True
   • False

   Hint

   Recall the definition of consistency.

5. 5.

   Given A = [[1, 2], [3, 6]] and b = [4, 8], find the ranks of A and [A|b], and determine the existence of a solution.

   Hint

   Calculate the ranks step by step.

6. 6.

   Create a system of equations where Rank(A) < Rank([A|b]) and explain why no solutions exist geometrically.

   Hint

   Visualize the equations on a graph.