25.4. Conditions for Uniqueness of Solution

Try these first

1. 1.

   True or False: A square matrix can have a unique solution.

   Hint

   Think about the properties that could ensure a unique solution.

2. 2.

   Define full column rank in your own words.

   Hint

   Consider how the rank relates to the number of columns.

3. 3.

   What condition must be satisfied for a linear system Ax = b to have a unique solution?

   • Rank(A) < n
   • Rank(A) = n
   • Rank(A) > n

   Hint

   Consider the implications of having fewer or more equations than unknowns.

4. 4.

   True or False: A rectangular matrix can, in some cases, provide a unique solution.

   • True
   • False

   Hint

   Think about how the matrix's rank relates to the system's variables.

5. 5.

   Given the following augmented matrix: [[1, 2, 3, 5], [2, 4, 6, 10]], analyze its rank and determine if the system is consistent and whether it has a unique solution.

   Hint

   Inspect the rows for linear dependency.

6. 6.

   For a system defined by a 4x4 matrix where three rows are linearly independent, determine what types of solutions are possible given three variables.

   Hint

   Think about how many dimensions are represented versus how much freedom there is in choosing solutions.