Question

n x n matrix A, where n >= 3. Select 3 statements from the invertible matrix...

n x n matrix A, where n >= 3. Select 3 statements from the invertible matrix theorem below and show that all 3 statements are true or false. Make sure to clearly explain and justify your work.

A=

-1 , 7, 9

7 , 7, 10

-3, -6, -4

  1. The equation A has only the trivial solution.

5. The columns of A form a linearly independent set.

6. The linear transformation x → Ax is one-to-one.

7. The equation Ax = b has at least one solution for each b ∈ Rn .

8. The columns of A span Rn .

9. The linear transformation maps Rn onto Rn .

10. There is an n × n matrix C such that CA = I.

11. There is an n × n matrix D such that AD = I.

12. AT is invertible.

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