Graham Haynes
05/13/2023 · Elementary School
Find the following matrices where \( A=\left[\begin{array}{cc}6 & -7 \\ -4 & 6 \\ -1 & 4\end{array}\right] \) and \( B=\left[\begin{array}{cc}-9 & 8 \\ 3 & 0 \\ -3 & 8\end{array}\right] \) O. A. \( -3 A-9 B=\square \) simplify your answ \( 0 B \). This matrix operation is not possible
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Tutor-Verified Answer
Step-by-step Solution
To find the matrix \( -3A - 9B \), we first need to calculate \( -3A \) and \( -9B \) separately.
Given:
\[
A = \begin{bmatrix} 6 & -7 \\ -4 & 6 \\ -1 & 4 \end{bmatrix}
\]
\[
B = \begin{bmatrix} -9 & 8 \\ 3 & 0 \\ -3 & 8 \end{bmatrix}
\]
### Step 1: Calculate \( -3A \)
\[
-3A = -3 \begin{bmatrix} 6 & -7 \\ -4 & 6 \\ -1 & 4 \end{bmatrix} = \begin{bmatrix} -3 \cdot 6 & -3 \cdot -7 \\ -3 \cdot -4 & -3 \cdot 6 \\ -3 \cdot -1 & -3 \cdot 4 \end{bmatrix}
\]
\[
= \begin{bmatrix} -18 & 21 \\ 12 & -18 \\ 3 & -12 \end{bmatrix}
\]
### Step 2: Calculate \( -9B \)
\[
-9B = -9 \begin{bmatrix} -9 & 8 \\ 3 & 0 \\ -3 & 8 \end{bmatrix} = \begin{bmatrix} -9 \cdot -9 & -9 \cdot 8 \\ -9 \cdot 3 & -9 \cdot 0 \\ -9 \cdot -3 & -9 \cdot 8 \end{bmatrix}
\]
\[
= \begin{bmatrix} 81 & -72 \\ -27 & 0 \\ 27 & -72 \end{bmatrix}
\]
### Step 3: Add \( -3A \) and \( -9B \)
Now we can add the two matrices \( -3A \) and \( -9B \):
\[
-3A - 9B = \begin{bmatrix} -18 & 21 \\ 12 & -18 \\ 3 & -12 \end{bmatrix} + \begin{bmatrix} 81 & -72 \\ -27 & 0 \\ 27 & -72 \end{bmatrix}
\]
We perform the addition element-wise:
\[
= \begin{bmatrix} -18 + 81 & 21 - 72 \\ 12 - 27 & -18 + 0 \\ 3 + 27 & -12 - 72 \end{bmatrix}
\]
\[
= \begin{bmatrix} 63 & -51 \\ -15 & -18 \\ 30 & -84 \end{bmatrix}
\]
### Final Result
Thus, the simplified result of \( -3A - 9B \) is:
\[
\boxed{\begin{bmatrix} 63 & -51 \\ -15 & -18 \\ 30 & -84 \end{bmatrix}}
\]
Quick Answer
\[
\boxed{\begin{bmatrix} 63 & -51 \\ -15 & -18 \\ 30 & -84 \end{bmatrix}}
\]
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