25.12. Singular and Ill-Conditioned Systems

Try these first

1. 1.

   What does it mean when a matrix is singular?

   Hint

   Consider the properties of determinants.

2. 2.

   How is the condition number of a matrix calculated?

   Hint

   Think about what happens to a matrix and its inverse.

3. 3.

   What indicates that a matrix is singular?

   • det(A) = 1
   • det(A) = 0
   • det(A) < 0

   Hint

   Think about how a determinant reflects the properties of a matrix.

4. 4.

   True or False: An ill-conditioned system has a condition number close to 1.

   • True
   • False

   Hint

   Reflect on the meaning of sensitivity in systems.

5. 5.

   Given the matrix A = [[1, 2], [2, 4]], determine if A is singular. Why or why not?

   Hint

   Consider the relationship between the rows or columns.

6. 6.

   If κ(A) = 50, explain what this implies about the stability of a linear system using A.

   Hint

   Reflect on external factors that could affect the model.