In Exercises 11 and 12, the matrices are all n × n. Each part ofthe 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" happensto be true. An implication is False if there is an instance inwhich "statement 2" is false but "statement 1" is true. Justify eachanswer.11. a. If the equation Ax = 0 has only the trivial solution, thenA is row equivalent to the n - n identity matrix.b. If the columns of A span R", then the columns are linearlyindependent.c. If A is an nx n matrix, then the equation Ax = b has atleast one solution for each b in R".d. If the equation Ax = 0 has a nontrivial solution, then Ahas fewer than n pivot positions.e. If AT is not invertible, then A is not invertible.

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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 R", then the columns are linearly independent. c. If A is an nx n matrix, then the equation Ax = b has at least one solution for each b in R".

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 R", then the columns are linearly independent. c. If A is an nx n matrix, then the equation Ax = b has at least one solution for each b in R".

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter6: Matrices And Determinants

Section: Chapter Questions

Problem 4P

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Question

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 R", then the columns are linearly
independent.
c. If A is an nx n matrix, then the equation Ax = b has at
least one solution for each b in R".
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.

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