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

Chapter 2, Problem 12E

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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 there is an n × n matrix D such that AD = I , then DA = I.b. If the linear transformation x ? Ax maps ?n into ?n, then the row reduced echelon form of A is I .c. If the columns of A are linearly independent, then the columns of A span ?n.d. If the equation Ax = b has at least one solution for each b in ?n, then the transformation x ? Ax is not one-to-one.e. If there is a b in ?n such that the equation Ax = b is consistent, then the solution is unique.