6. 12. 0, 0.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.2: Guassian Elimination And Matrix Methods
Problem 94E
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1.2 number 12 review
![### Linear Algebra: Row Reduction and Solutions of Linear Systems
#### Exercises 3 and 4
**Objective:** Row reduce the matrices in Exercises 3 and 4 and convert them to reduced echelon form. Circle the pivot positions in the final matrix, and in the original matrix, and list the pivot columns.
3.
\[
\begin{bmatrix}
1 & 2 & 4 & 8 \\
2 & 4 & 6 & 8 \\
1 & 6 & 9 & 12 \\
\end{bmatrix}
\]
4.
\[
\begin{bmatrix}
1 & 2 & 4 & 5 \\
2 & 4 & 5 & 4 \\
4 & 5 & 4 & 2 \\
\end{bmatrix}
\]
#### Exercise 5
**Objective:** Describe the possible echelon forms of a nonzero \(2 \times 2\) matrix. Use the symbols \(\ast\), \(\diamond\), and 0, as in the first part of Example 1.
#### Exercise 6
**Objective:** Repeat Exercise 5 for a nonzero \(3 \times 2\) matrix.
#### Exercises 7-14
**Objective:** Find the general solutions of the systems whose augmented matrices are given.
7.
\[
\begin{bmatrix}
1 & 3 & 4 & 7 \\
3 & 9 & 7 & 6 \\
\end{bmatrix}
\]
8.
\[
\begin{bmatrix}
1 & -3 & 0 & 5 \\
3 & 7 & 0 & 9 \\
\end{bmatrix}
\]
9.
\[
\begin{bmatrix}
1 & 1 & 3 & -2 \\
1 & 2 & 5 & -4 \\
\end{bmatrix}
\]
10.
\[
\begin{bmatrix}
1 & 0 & 3 & 0 \\
2 & -2 & 5 & 4 \\
\end{bmatrix}
\]
11.
\[
\begin{bmatrix}
3 & -2 & 4 & 0 \\
6 & -4 & 8 & 0 \\
\end{bmatrix}
\]
12.
\[
\begin{bmatrix}
2 & 6](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2Fbed17c1e-cf6c-4467-ac3e-919b65a16570%2Fm75n7ly.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra: Row Reduction and Solutions of Linear Systems
#### Exercises 3 and 4
**Objective:** Row reduce the matrices in Exercises 3 and 4 and convert them to reduced echelon form. Circle the pivot positions in the final matrix, and in the original matrix, and list the pivot columns.
3.
\[
\begin{bmatrix}
1 & 2 & 4 & 8 \\
2 & 4 & 6 & 8 \\
1 & 6 & 9 & 12 \\
\end{bmatrix}
\]
4.
\[
\begin{bmatrix}
1 & 2 & 4 & 5 \\
2 & 4 & 5 & 4 \\
4 & 5 & 4 & 2 \\
\end{bmatrix}
\]
#### Exercise 5
**Objective:** Describe the possible echelon forms of a nonzero \(2 \times 2\) matrix. Use the symbols \(\ast\), \(\diamond\), and 0, as in the first part of Example 1.
#### Exercise 6
**Objective:** Repeat Exercise 5 for a nonzero \(3 \times 2\) matrix.
#### Exercises 7-14
**Objective:** Find the general solutions of the systems whose augmented matrices are given.
7.
\[
\begin{bmatrix}
1 & 3 & 4 & 7 \\
3 & 9 & 7 & 6 \\
\end{bmatrix}
\]
8.
\[
\begin{bmatrix}
1 & -3 & 0 & 5 \\
3 & 7 & 0 & 9 \\
\end{bmatrix}
\]
9.
\[
\begin{bmatrix}
1 & 1 & 3 & -2 \\
1 & 2 & 5 & -4 \\
\end{bmatrix}
\]
10.
\[
\begin{bmatrix}
1 & 0 & 3 & 0 \\
2 & -2 & 5 & 4 \\
\end{bmatrix}
\]
11.
\[
\begin{bmatrix}
3 & -2 & 4 & 0 \\
6 & -4 & 8 & 0 \\
\end{bmatrix}
\]
12.
\[
\begin{bmatrix}
2 & 6
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