Chapter7: Systems Of Equations And Inequalities
Section7.6: Solving Systems With Gaussian Elimination
Problem 5SE: Can a matrix that has 0 entries for an entire row have one solution? Explain why or why not.
Related questions
Question
![**Problem: Find the transpose of the matrix.**
Given Matrix:
\[
\begin{bmatrix}
8 & 4 \\
-4 & 0 \\
-7 & 7 \\
\end{bmatrix}
\]
**Options:**
1. \[
\begin{bmatrix}
4 & 8 \\
0 & -4 \\
7 & -7 \\
\end{bmatrix}
\]
2. \[
\begin{bmatrix}
4 & 0 & 7 \\
8 & -4 & -7 \\
\end{bmatrix}
\]
3. \[
\begin{bmatrix}
8 & -4 & -7 \\
4 & 0 & 7 \\
\end{bmatrix}
\]
4. \[
\begin{bmatrix}
-7 & 7 \\
-4 & 0 \\
8 & 4 \\
\end{bmatrix}
\]
**Explanation:**
The transpose of a matrix is formed by swapping the rows and columns. For the given matrix:
- Row 1: \([8, 4]\) becomes Column 1: \([8, -4, -7]\)
- Row 2: \([-4, 0]\) becomes Column 2: \([4, 0, 7]\)
Therefore, the correct transpose is Option 3:
\[
\begin{bmatrix}
8 & -4 & -7 \\
4 & 0 & 7 \\
\end{bmatrix}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac0f3c56-5561-4b47-8ddf-f02546f51c3e%2F69041c66-3ea0-4bd8-b6e5-6bcc51e2972b%2Fm9rmhg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem: Find the transpose of the matrix.**
Given Matrix:
\[
\begin{bmatrix}
8 & 4 \\
-4 & 0 \\
-7 & 7 \\
\end{bmatrix}
\]
**Options:**
1. \[
\begin{bmatrix}
4 & 8 \\
0 & -4 \\
7 & -7 \\
\end{bmatrix}
\]
2. \[
\begin{bmatrix}
4 & 0 & 7 \\
8 & -4 & -7 \\
\end{bmatrix}
\]
3. \[
\begin{bmatrix}
8 & -4 & -7 \\
4 & 0 & 7 \\
\end{bmatrix}
\]
4. \[
\begin{bmatrix}
-7 & 7 \\
-4 & 0 \\
8 & 4 \\
\end{bmatrix}
\]
**Explanation:**
The transpose of a matrix is formed by swapping the rows and columns. For the given matrix:
- Row 1: \([8, 4]\) becomes Column 1: \([8, -4, -7]\)
- Row 2: \([-4, 0]\) becomes Column 2: \([4, 0, 7]\)
Therefore, the correct transpose is Option 3:
\[
\begin{bmatrix}
8 & -4 & -7 \\
4 & 0 & 7 \\
\end{bmatrix}
\]
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