Solved: In Exercises 11 and 12, the matrices are all n ×

Chapter 2, Problem 11E

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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.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 ?n, 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 ?n.d. If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivot positions.e. If AT is not invertible, then A is not invertible.