21. Let Ax = 0 be a homogeneous system of n linear equations in n unknowns that has only the trivial solution. Prove that if k is any positive integer, then the system Akx = 0 also has only the trivial solution.
21. Let Ax = 0 be a homogeneous system of n linear equations in n unknowns that has only the trivial solution. Prove that if k is any positive integer, then the system Akx = 0 also has only the trivial solution.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.3: The Inverse Of A Matrix
Problem 20EQ: In Exercises 20-23, solve the given matrix equation for X. Simplify your answers as much as...
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Question
F.16 question 21 on paper please
![2
[x₁
A =
2
2
-2
and
--B
X = X₂
3
1
1
a. Show that the equation
Ax= x can be rewritten as
(A - I)x= 0 and use this result to solve Ax = x for x.
b. Solve Ax = 4x.
In Exercises 19-20, solve the matrix equation for X.
1
-1
1
2
5
7
8
19. 2
3
0X = 4 0
-3
0
1
0
2
3 5
2
1
-2
4
3 2 1
20.
-1 X = 6
7 8 9
1
1 -4]
1
3 7 9]
Working with Proofs
21. Let Ax = 0 be a homogeneous system of n linear equations in
n unknowns that has only the trivial solution. Prove that if k
is any positive integer, then the system Akx = 0 also has only
the trivial solution.
T
22. Let Ax = 0 be a homogeneous system of n linear equations
in n unknowns, and let Q be an invertible nxn matrix.
Prove that Ax = 0 has only the trivial solution if and only if
(QA)x= 0 has only the trivial solution.
23. Let Ax=b be any consistent system of linear equations, and
let x₁ be a fixed solution. Prove that every solution to the sys-
0 1]
-1
Wa
T1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccdd90b5-cd41-4340-8a36-15b102c9ba73%2F02c65e08-1f75-4e53-8753-254d508e253d%2Fqrpxq6l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2
[x₁
A =
2
2
-2
and
--B
X = X₂
3
1
1
a. Show that the equation
Ax= x can be rewritten as
(A - I)x= 0 and use this result to solve Ax = x for x.
b. Solve Ax = 4x.
In Exercises 19-20, solve the matrix equation for X.
1
-1
1
2
5
7
8
19. 2
3
0X = 4 0
-3
0
1
0
2
3 5
2
1
-2
4
3 2 1
20.
-1 X = 6
7 8 9
1
1 -4]
1
3 7 9]
Working with Proofs
21. Let Ax = 0 be a homogeneous system of n linear equations in
n unknowns that has only the trivial solution. Prove that if k
is any positive integer, then the system Akx = 0 also has only
the trivial solution.
T
22. Let Ax = 0 be a homogeneous system of n linear equations
in n unknowns, and let Q be an invertible nxn matrix.
Prove that Ax = 0 has only the trivial solution if and only if
(QA)x= 0 has only the trivial solution.
23. Let Ax=b be any consistent system of linear equations, and
let x₁ be a fixed solution. Prove that every solution to the sys-
0 1]
-1
Wa
T1.
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