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.
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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Ff1.6 question 21 on paper please
![### Solving Matrix Equations
#### In Exercises 19–20, solve the matrix equation for \( X \).
19. \[
\begin{pmatrix}
1 & -1 & 1 \\
2 & 3 & 0 \\
0 & 2 & -1
\end{pmatrix}
X =
\begin{pmatrix}
2 & -1 & 5 & 7 & 8 \\
4 & 0 & -3 & 0 & 1 \\
3 & 5 & -7 & 2 & 1
\end{pmatrix}
\]
20. \[
\begin{pmatrix}
-2 & 0 & 1 \\
0 & -1 & -1 \\
1 & 1 & -4
\end{pmatrix}
X =
\begin{pmatrix}
4 & 3 & 2 & 1 \\
6 & 7 & 8 & 9 \\
1 & 3 & 7 & 9
\end{pmatrix}
\]
#### Working with Proofs
21. Let \( A\mathbf{x} = 0 \) 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 \( A^{k}\mathbf{x} = 0 \) also has only the trivial solution.
22. Let \( A\mathbf{x} = 0 \) be a homogeneous system of \( n \) linear equations in \( n \) unknowns, and let \( Q \) be an invertible \( n \times n \) matrix. Prove that \( A\mathbf{x} = 0 \) has only the trivial solution if and only if \( (QA)\mathbf{x} = 0 \) has only the trivial solution.
23. Let \( A\mathbf{x} = \mathbf{b} \) be any consistent system of linear equations, and let \( \mathbf{x_1} \) be a fixed solution. Prove that every solution to the system can be written as \( \mathbf{x} = \mathbf{x_1} + \mathbf{x_2} \), where \( \mathbf{x_2} \) is a solution to the corresponding homogeneous system \( A\mathbf{x} = 0 \).
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Transcribed Image Text:### Solving Matrix Equations
#### In Exercises 19–20, solve the matrix equation for \( X \).
19. \[
\begin{pmatrix}
1 & -1 & 1 \\
2 & 3 & 0 \\
0 & 2 & -1
\end{pmatrix}
X =
\begin{pmatrix}
2 & -1 & 5 & 7 & 8 \\
4 & 0 & -3 & 0 & 1 \\
3 & 5 & -7 & 2 & 1
\end{pmatrix}
\]
20. \[
\begin{pmatrix}
-2 & 0 & 1 \\
0 & -1 & -1 \\
1 & 1 & -4
\end{pmatrix}
X =
\begin{pmatrix}
4 & 3 & 2 & 1 \\
6 & 7 & 8 & 9 \\
1 & 3 & 7 & 9
\end{pmatrix}
\]
#### Working with Proofs
21. Let \( A\mathbf{x} = 0 \) 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 \( A^{k}\mathbf{x} = 0 \) also has only the trivial solution.
22. Let \( A\mathbf{x} = 0 \) be a homogeneous system of \( n \) linear equations in \( n \) unknowns, and let \( Q \) be an invertible \( n \times n \) matrix. Prove that \( A\mathbf{x} = 0 \) has only the trivial solution if and only if \( (QA)\mathbf{x} = 0 \) has only the trivial solution.
23. Let \( A\mathbf{x} = \mathbf{b} \) be any consistent system of linear equations, and let \( \mathbf{x_1} \) be a fixed solution. Prove that every solution to the system can be written as \( \mathbf{x} = \mathbf{x_1} + \mathbf{x_2} \), where \( \mathbf{x_2} \) is a solution to the corresponding homogeneous system \( A\mathbf{x} = 0 \).
---
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