Determine if the columns of the matrix form a linearly independent set. Justify your answer. Select the correct choice below and fill in the answer box within your choice. (Type an integer or simplified fraction for each matrix element.) O A. If A is the given matrix, then the augmented matrix -4 -3 0-1 1 2 1 1 reduced echelon form of this matrix indicates that Ax = 0 has only the trivial solution. Therefore, the columns of A form a linearly independent set. 0 6 -6 - 12 represents the equation Ax = 0. The O B. IfA is the given matrix, then the augmented matrix represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax=0 has more than one solution. Therefore, the columns of A form a linearly independent set. OC. If A is the given matrix, then the augmented matrix represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax=0 has more than one solution. Therefore, the columns of A do not form a linearly independent set. O D. If A is the given matrix, then the augmented matrix represents the equation Ax=0. The reduced echelon form of this matrix indicates that Ax=0 has only the trivial solution. Therefore, the columns of A
Determine if the columns of the matrix form a linearly independent set. Justify your answer. Select the correct choice below and fill in the answer box within your choice. (Type an integer or simplified fraction for each matrix element.) O A. If A is the given matrix, then the augmented matrix -4 -3 0-1 1 2 1 1 reduced echelon form of this matrix indicates that Ax = 0 has only the trivial solution. Therefore, the columns of A form a linearly independent set. 0 6 -6 - 12 represents the equation Ax = 0. The O B. IfA is the given matrix, then the augmented matrix represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax=0 has more than one solution. Therefore, the columns of A form a linearly independent set. OC. If A is the given matrix, then the augmented matrix represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax=0 has more than one solution. Therefore, the columns of A do not form a linearly independent set. O D. If A is the given matrix, then the augmented matrix represents the equation Ax=0. The reduced echelon form of this matrix indicates that Ax=0 has only the trivial solution. Therefore, the columns of A
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
![### Linear Independence of Matrix Columns
#### Determine if the columns of the matrix form a linearly independent set. Justify your answer.
Given Matrix:
\[
\begin{pmatrix}
-4 & -3 & 0 \\
0 & -1 & 6 \\
1 & 1 & -6 \\
2 & 1 & -12 \\
\end{pmatrix}
\]
---
#### Explanation:
To determine if the columns of the matrix form a linearly independent set, we need to ascertain whether the equation \(Ax = 0\) has only the trivial solution (i.e., \(x = 0\)).
---
#### Multiple Choice Question:
Select the correct choice below and fill in the answer box within your choice.
(Type an integer or simplified fraction for each matrix element.)
---
**A.**
If \(A\) is the given matrix, then the augmented matrix
\[
\begin{pmatrix}
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\end{pmatrix}
\]
represents the equation \(Ax = 0\). The reduced echelon form of this matrix indicates that \(Ax = 0\) has only the trivial solution. Therefore, the columns of \(A\) form a linearly independent set.
---
**B.**
If \(A\) is the given matrix, then the augmented matrix
\[
\begin{pmatrix}
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\end{pmatrix}
\]
represents the equation \(Ax = 0\). The reduced echelon form of this matrix indicates that \(Ax = 0\) has more than one solution. Therefore, the columns of \(A\) do not form a linearly independent set.
---
**C.**
If \(A\) is the given matrix, then the augmented matrix
\[
\begin{pmatrix}
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8109dbb5-39b2-442d-b17d-793e299fca4a%2Ffe92dd69-8557-4c3e-af02-df10476f5147%2Fg2l9zrt_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Independence of Matrix Columns
#### Determine if the columns of the matrix form a linearly independent set. Justify your answer.
Given Matrix:
\[
\begin{pmatrix}
-4 & -3 & 0 \\
0 & -1 & 6 \\
1 & 1 & -6 \\
2 & 1 & -12 \\
\end{pmatrix}
\]
---
#### Explanation:
To determine if the columns of the matrix form a linearly independent set, we need to ascertain whether the equation \(Ax = 0\) has only the trivial solution (i.e., \(x = 0\)).
---
#### Multiple Choice Question:
Select the correct choice below and fill in the answer box within your choice.
(Type an integer or simplified fraction for each matrix element.)
---
**A.**
If \(A\) is the given matrix, then the augmented matrix
\[
\begin{pmatrix}
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\end{pmatrix}
\]
represents the equation \(Ax = 0\). The reduced echelon form of this matrix indicates that \(Ax = 0\) has only the trivial solution. Therefore, the columns of \(A\) form a linearly independent set.
---
**B.**
If \(A\) is the given matrix, then the augmented matrix
\[
\begin{pmatrix}
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\end{pmatrix}
\]
represents the equation \(Ax = 0\). The reduced echelon form of this matrix indicates that \(Ax = 0\) has more than one solution. Therefore, the columns of \(A\) do not form a linearly independent set.
---
**C.**
If \(A\) is the given matrix, then the augmented matrix
\[
\begin{pmatrix}
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed
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