Chapter 2.3, Problem 12E

In Exercises 11 and 12, the matrices are all n × n. Each part of the exercises is an implication of the form "If “statement 1", then “statement 2”. Mark an implication as True if the truth of “statement 2” always follows whenever “statement 1” happens to be true. An implication is False if there is an instance in which “statement 2” is false but “statement 1” is true. Justify each answer.

12. a. If there is an n × n matrix D such that AD = I, then there is also an n × n matrix C such that CA = I.

b. If the columns of A are linearly independent, then the columns of A span R".

c. If the equation Ax = b has at least one solution for each b in R", then the solution is unique for each b.

d. If the linear transformation (x) ↦ Ax maps ℝⁿ into ℝⁿ, then A has n pivot positions.

e. If there is a b in ℝⁿ such that the equation Ax = b is inconsistent, then the transformation x ↦ Ax is not one-to-one.