Santiago Stephens
01/09/2023 · Elementary School
Find the following matrices where \( A=\left[\begin{array}{rrr}-13 & 11 & 5 \\ 7 & -13 & -1 \\ 1 & -14 & -7\end{array}\right] \) and \( B=\left[\begin{array}{rrr}5 & 0 & 8 \\ 0 & 11 & 3 \\ 6 & 6 & -13\end{array}\right] \) a. \( A+B \quad \) b. \( A-B \quad \) d. \( 5 A-5 B \) a. Select the correct choice below and if necessary, fal in the answer box to complete your oh A. \( A+B= \) (Simplity your answer) B. This matrix operation is not possible
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Tutor-Verified Answer
Step-by-step Solution
To perform the matrix operations, we will add and subtract the matrices \( A \) and \( B \) as well as calculate \( 5A - 5B \).
Given:
\[
A = \begin{bmatrix}
-13 & 11 & 5 \\
7 & -13 & -1 \\
1 & -14 & -7
\end{bmatrix}
\]
\[
B = \begin{bmatrix}
5 & 0 & 8 \\
0 & 11 & 3 \\
6 & 6 & -13
\end{bmatrix}
\]
### a. \( A + B \)
To add the matrices, we add the corresponding elements:
\[
A + B = \begin{bmatrix}
-13 + 5 & 11 + 0 & 5 + 8 \\
7 + 0 & -13 + 11 & -1 + 3 \\
1 + 6 & -14 + 6 & -7 - 13
\end{bmatrix}
\]
Calculating each element:
\[
A + B = \begin{bmatrix}
-8 & 11 & 13 \\
7 & -2 & 2 \\
7 & -8 & -20
\end{bmatrix}
\]
### b. \( A - B \)
To subtract the matrices, we subtract the corresponding elements:
\[
A - B = \begin{bmatrix}
-13 - 5 & 11 - 0 & 5 - 8 \\
7 - 0 & -13 - 11 & -1 - 3 \\
1 - 6 & -14 - 6 & -7 + 13
\end{bmatrix}
\]
Calculating each element:
\[
A - B = \begin{bmatrix}
-18 & 11 & -3 \\
7 & -24 & -4 \\
-5 & -20 & 6
\end{bmatrix}
\]
### d. \( 5A - 5B \)
First, we calculate \( 5A \) and \( 5B \):
\[
5A = 5 \cdot \begin{bmatrix}
-13 & 11 & 5 \\
7 & -13 & -1 \\
1 & -14 & -7
\end{bmatrix} = \begin{bmatrix}
-65 & 55 & 25 \\
35 & -65 & -5 \\
5 & -70 & -35
\end{bmatrix}
\]
\[
5B = 5 \cdot \begin{bmatrix}
5 & 0 & 8 \\
0 & 11 & 3 \\
6 & 6 & -13
\end{bmatrix} = \begin{bmatrix}
25 & 0 & 40 \\
0 & 55 & 15 \\
30 & 30 & -65
\end{bmatrix}
\]
Now, we can calculate \( 5A - 5B \):
\[
5A - 5B = \begin{bmatrix}
-65 - 25 & 55 - 0 & 25 - 40 \\
35 - 0 & -65 - 55 & -5 - 15 \\
5 - 30 & -70 - 30 & -35 + 65
\end{bmatrix}
\]
Calculating each element:
\[
5A - 5B = \begin{bmatrix}
-90 & 55 & -15 \\
35 & -120 & -20 \\
-25 & -100 & 30
\end{bmatrix}
\]
### Summary of Results
a. \( A + B = \begin{bmatrix} -8 & 11 & 13 \\ 7 & -2 & 2 \\ 7 & -8 & -20 \end{bmatrix} \)
b. \( A - B = \begin{bmatrix} -18 & 11 & -3 \\ 7 & -24 & -4 \\ -5 & -20 & 6 \end{bmatrix} \)
d. \( 5A - 5B = \begin{bmatrix} -90 & 55 & -15 \\ 35 & -120 & -20 \\ -25 & -100 & 30 \end{bmatrix} \)
Quick Answer
a. \( A + B = \begin{bmatrix} -8 & 11 & 13 \\ 7 & -2 & 2 \\ 7 & -8 & -20 \end{bmatrix} \)
b. \( A - B = \begin{bmatrix} -18 & 11 & -3 \\ 7 & -24 & -4 \\ -5 & -20 & 6 \end{bmatrix} \)
d. \( 5A - 5B = \begin{bmatrix} -90 & 55 & -15 \\ 35 & -120 & -20 \\ -25 & -100 & 30 \end{bmatrix} \)
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