Chapter7: Systems Of Equations And Inequalities
Section7.6: Solving Systems With Gaussian Elimination
Problem 1SE: Can any system of linear equations be written as an augmented matrix? Explain why or why not....
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![**Problem Statement:**
If \( A + B = C \), use the matrices below to find the element \( C_{1,2} \)?
\[ A = \begin{bmatrix}
2 & -3 \\
-1 & 4
\end{bmatrix} \]
\[ B = \begin{bmatrix}
4 & -5 \\
1 & 3
\end{bmatrix} \]
---
**Solution:**
To find \( C_{1,2} \), we need to compute the matrix \( C \) by adding the corresponding elements of matrices \( A \) and \( B \).
Given:
\[ A = \begin{bmatrix}
2 & -3 \\
-1 & 4
\end{bmatrix} \]
\[ B = \begin{bmatrix}
4 & -5 \\
1 & 3
\end{bmatrix} \]
**Step-by-Step Calculation:**
1. Add the elements in the first row and first column:
\[ C_{1,1} = A_{1,1} + B_{1,1} = 2 + 4 = 6 \]
2. Add the elements in the first row and second column:
\[ C_{1,2} = A_{1,2} + B_{1,2} = -3 + (-5) = -8 \]
3. Add the elements in the second row and first column:
\[ C_{2,1} = A_{2,1} + B_{2,1} = -1 + 1 = 0 \]
4. Add the elements in the second row and second column:
\[ C_{2,2} = A_{2,2} + B_{2,2} = 4 + 3 = 7 \]
Thus, the matrix \( C \) is:
\[ C = \begin{bmatrix}
6 & -8 \\
0 & 7
\end{bmatrix} \]
Therefore, the element \( C_{1,2} \) is \( -8 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5bbc26c4-f789-4881-a3b2-dd815a2e0938%2Fa6c41c80-f108-4b3f-9483-621af5d16054%2F93nenz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
If \( A + B = C \), use the matrices below to find the element \( C_{1,2} \)?
\[ A = \begin{bmatrix}
2 & -3 \\
-1 & 4
\end{bmatrix} \]
\[ B = \begin{bmatrix}
4 & -5 \\
1 & 3
\end{bmatrix} \]
---
**Solution:**
To find \( C_{1,2} \), we need to compute the matrix \( C \) by adding the corresponding elements of matrices \( A \) and \( B \).
Given:
\[ A = \begin{bmatrix}
2 & -3 \\
-1 & 4
\end{bmatrix} \]
\[ B = \begin{bmatrix}
4 & -5 \\
1 & 3
\end{bmatrix} \]
**Step-by-Step Calculation:**
1. Add the elements in the first row and first column:
\[ C_{1,1} = A_{1,1} + B_{1,1} = 2 + 4 = 6 \]
2. Add the elements in the first row and second column:
\[ C_{1,2} = A_{1,2} + B_{1,2} = -3 + (-5) = -8 \]
3. Add the elements in the second row and first column:
\[ C_{2,1} = A_{2,1} + B_{2,1} = -1 + 1 = 0 \]
4. Add the elements in the second row and second column:
\[ C_{2,2} = A_{2,2} + B_{2,2} = 4 + 3 = 7 \]
Thus, the matrix \( C \) is:
\[ C = \begin{bmatrix}
6 & -8 \\
0 & 7
\end{bmatrix} \]
Therefore, the element \( C_{1,2} \) is \( -8 \).
Transcribed Image Text:**Understanding Matrix Dimensions**
When writing the dimensions of a matrix with 6 columns and 5 rows, one should list the number of columns first (i.e., 6 x 5).
**Question:**
Select one:
- O True
- O False
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