In Exercises 11 and 12, the matrices are all nxn. 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 I A is row equivalent to the n x n identity matrix. b. If the columns of A span R", then the columns are linearly independent. c. If A is an n x 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. 12. a. If there is an n x n matrix D such that AD = I, then there is also an n x 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.
In Exercises 11 and 12, the matrices are all nxn. 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 I A is row equivalent to the n x n identity matrix. b. If the columns of A span R", then the columns are linearly independent. c. If A is an n x 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. 12. a. If there is an n x n matrix D such that AD = I, then there is also an n x 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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Need ans to 12 only pls
Transcribed Image Text:d. If the linear transformation (x) → Ax maps R" into R",
then A has n pivot positions.
e. If there is a b in R" such that the equation Ax = b is
inconsistent, then the transformation x → Ax is not one-
to-one.
13. An mxn upper triangular matrix is one whose entries
below the main diagonal are O's (as in Exercise 8). When
is a square upper triangular matrix invertible? Justify your
answer.
I
14. An mxn lower triangular matrix is one whose entries
above the main diagonal are 0's (as in Exercise 3). When
is a square lower triangular matrix invertible? Justify your
answer.
15. Can a square matrix with two identical columns be invert-
ible? Why or why not?
Transcribed Image Text:In Exercises 11 and 12, the matrices are all nxn. 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 I
A is row equivalent to the n x n identity matrix.
b. If the columns of A span R", then the columns are linearly
independent.
c. If A is an n x 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 A¹ is not invertible, then A is not invertible.
12. a. If there is an n x n matrix D such that AD = I, then there
is also an n x 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.
20. If n x n matrices E and
then E and F commute.
21. If the equation Gx = y
y in R", can the column
22. If the equation Hx = ci
can you say about the ec
If an n x n matrix K can
you say about the colum
If L is n x n and the equa
do the columns of L span
23.
24.
25. Verify the boxed stateme
26. Explain why the colum
columns of A are linearly
27. Show that if AB is invertib
6(b), because you cannot
[Hint: There is a matrix F
28. Show that if AB is inverti
29. If A is an n x n matrix and
one solution for some b,
not one-to-one. What else
tion? Justify your answer.
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