Chapter 2.3, Problem 11E

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.

11. a. If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n × n identity matrix.

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

c. If A is an n × n matrix, then the equation, Ax = b has at least one solution for each b in ℝⁿ.

d. If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivot positions.

e. If A^T is not invertible, then A is not invertible.