24. Suppose A is n x n and the equation Ax = b has a solution for each b in R". Explain why A must be invertible. [Hint: Is A row equivalent to I?]
24. Suppose A is n x n and the equation Ax = b has a solution for each b in R". Explain why A must be invertible. [Hint: Is A row equivalent to I?]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Ans 24
![trivial solution. Explain why A has n pivot columns and A is
row equivalent to I. By Theorem 7, this shows that A must
be invertible. (This exercise and Exercise 24 will be cited in
Section 2.3.)
24. Suppose A is n x n and the equation Ax = b has a solution
for each b in R". Explain why A must be invertible. [Hint: Is
A row equivalent to In?]
Exercises 25 and 26 prove Theorem 4 for A
[ ]
d
25. Show that if ad bc = 0, then the equation Ax = 0 has
more than one solution. Why does this imply that A is not
invertible? [Hint: First, consider a = b = 0. Then, if a and
b
[2]
a
26. Show that if ad - bc #0, the formula for A works.
wom
b are not both zero, consider the vector x =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6147cd51-fae9-4b86-867f-e7a7fe5750a1%2Fe0dee704-2ce9-40dc-a72a-dafdf611e78c%2Fwg0oitt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:trivial solution. Explain why A has n pivot columns and A is
row equivalent to I. By Theorem 7, this shows that A must
be invertible. (This exercise and Exercise 24 will be cited in
Section 2.3.)
24. Suppose A is n x n and the equation Ax = b has a solution
for each b in R". Explain why A must be invertible. [Hint: Is
A row equivalent to In?]
Exercises 25 and 26 prove Theorem 4 for A
[ ]
d
25. Show that if ad bc = 0, then the equation Ax = 0 has
more than one solution. Why does this imply that A is not
invertible? [Hint: First, consider a = b = 0. Then, if a and
b
[2]
a
26. Show that if ad - bc #0, the formula for A works.
wom
b are not both zero, consider the vector x =
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