### Educational Content: Matrix Operations #### Problem Statement: Find \( (3)BA + (4)AC \), if possible. #### Given Matrices: \[ A = \begin{bmatrix} 2 & -3 & 3 \\ 0 & 1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} -1 & 1 \\ 1 & 0 \end{bmatrix}, \quad C = \begin{bmatrix} -1 & 0 & 2 \\ 0 & 1 & -1 \\ 0 & -1 & 1 \end{bmatrix} \] #### Solution Steps: A. Calculate \( (3)BA + (4)AC \): \[ (3)BA + (4)AC = \] (Placeholder for the result) B. (Alternative Outcome) \[ \text{The matrix is not defined.} \] #### Explanation: To determine if the expression \( (3)BA + (4)AC \) is defined, we must check the dimensions of the resulting matrices from each multiplication. 1. **Dimensions of \( B \) and \( A \):** - Matrix \( B \): 2 rows × 2 columns. - Matrix \( A \): 2 rows × 3 columns. - Multiplying \( B \) and \( A \) requires the number of columns in \( B \) to equal the number of rows in \( A \), which is not the case here (2 ≠ 2). So, \( BA \) is not defined. 2. **Dimensions of \( A \) and \( C \):** - Matrix \( A \): 2 rows × 3 columns. - Matrix \( C \): 3 rows × 3 columns. - Multiplying \( A \) and \( C \) is defined and results in a matrix with dimensions 2 × 3. Given that \( BA \) is not defined: \[ \text{The matrix is not defined.} \] ### Conclusion: For the given matrices \( A \), \( B \), and \( C \), the expression \( (3)BA + (4)AC \) cannot be computed because \( BA \) is not a defined operation based on the matrix dimensions provided.
### Educational Content: Matrix Operations #### Problem Statement: Find \( (3)BA + (4)AC \), if possible. #### Given Matrices: \[ A = \begin{bmatrix} 2 & -3 & 3 \\ 0 & 1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} -1 & 1 \\ 1 & 0 \end{bmatrix}, \quad C = \begin{bmatrix} -1 & 0 & 2 \\ 0 & 1 & -1 \\ 0 & -1 & 1 \end{bmatrix} \] #### Solution Steps: A. Calculate \( (3)BA + (4)AC \): \[ (3)BA + (4)AC = \] (Placeholder for the result) B. (Alternative Outcome) \[ \text{The matrix is not defined.} \] #### Explanation: To determine if the expression \( (3)BA + (4)AC \) is defined, we must check the dimensions of the resulting matrices from each multiplication. 1. **Dimensions of \( B \) and \( A \):** - Matrix \( B \): 2 rows × 2 columns. - Matrix \( A \): 2 rows × 3 columns. - Multiplying \( B \) and \( A \) requires the number of columns in \( B \) to equal the number of rows in \( A \), which is not the case here (2 ≠ 2). So, \( BA \) is not defined. 2. **Dimensions of \( A \) and \( C \):** - Matrix \( A \): 2 rows × 3 columns. - Matrix \( C \): 3 rows × 3 columns. - Multiplying \( A \) and \( C \) is defined and results in a matrix with dimensions 2 × 3. Given that \( BA \) is not defined: \[ \text{The matrix is not defined.} \] ### Conclusion: For the given matrices \( A \), \( B \), and \( C \), the expression \( (3)BA + (4)AC \) cannot be computed because \( BA \) is not a defined operation based on the matrix dimensions provided.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.3: Matrix Algebra
Problem 85E: Determine if the statement is true or false. If the statement is false, then correct it and make it...
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![### Educational Content: Matrix Operations
#### Problem Statement:
Find \( (3)BA + (4)AC \), if possible.
#### Given Matrices:
\[
A = \begin{bmatrix}
2 & -3 & 3 \\
0 & 1 & 0
\end{bmatrix}, \quad
B = \begin{bmatrix}
-1 & 1 \\
1 & 0
\end{bmatrix}, \quad
C = \begin{bmatrix}
-1 & 0 & 2 \\
0 & 1 & -1 \\
0 & -1 & 1
\end{bmatrix}
\]
#### Solution Steps:
A. Calculate \( (3)BA + (4)AC \):
\[
(3)BA + (4)AC =
\]
(Placeholder for the result)
B. (Alternative Outcome)
\[
\text{The matrix is not defined.}
\]
#### Explanation:
To determine if the expression \( (3)BA + (4)AC \) is defined, we must check the dimensions of the resulting matrices from each multiplication.
1. **Dimensions of \( B \) and \( A \):**
- Matrix \( B \): 2 rows × 2 columns.
- Matrix \( A \): 2 rows × 3 columns.
- Multiplying \( B \) and \( A \) requires the number of columns in \( B \) to equal the number of rows in \( A \), which is not the case here (2 ≠ 2). So, \( BA \) is not defined.
2. **Dimensions of \( A \) and \( C \):**
- Matrix \( A \): 2 rows × 3 columns.
- Matrix \( C \): 3 rows × 3 columns.
- Multiplying \( A \) and \( C \) is defined and results in a matrix with dimensions 2 × 3.
Given that \( BA \) is not defined:
\[
\text{The matrix is not defined.}
\]
### Conclusion:
For the given matrices \( A \), \( B \), and \( C \), the expression \( (3)BA + (4)AC \) cannot be computed because \( BA \) is not a defined operation based on the matrix dimensions provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd0cd4ae9-86e1-45b4-8824-a362b78dd084%2Fa4431c65-a5d9-4962-b6fc-1700ad5a3b58%2Fdsudwmo_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Content: Matrix Operations
#### Problem Statement:
Find \( (3)BA + (4)AC \), if possible.
#### Given Matrices:
\[
A = \begin{bmatrix}
2 & -3 & 3 \\
0 & 1 & 0
\end{bmatrix}, \quad
B = \begin{bmatrix}
-1 & 1 \\
1 & 0
\end{bmatrix}, \quad
C = \begin{bmatrix}
-1 & 0 & 2 \\
0 & 1 & -1 \\
0 & -1 & 1
\end{bmatrix}
\]
#### Solution Steps:
A. Calculate \( (3)BA + (4)AC \):
\[
(3)BA + (4)AC =
\]
(Placeholder for the result)
B. (Alternative Outcome)
\[
\text{The matrix is not defined.}
\]
#### Explanation:
To determine if the expression \( (3)BA + (4)AC \) is defined, we must check the dimensions of the resulting matrices from each multiplication.
1. **Dimensions of \( B \) and \( A \):**
- Matrix \( B \): 2 rows × 2 columns.
- Matrix \( A \): 2 rows × 3 columns.
- Multiplying \( B \) and \( A \) requires the number of columns in \( B \) to equal the number of rows in \( A \), which is not the case here (2 ≠ 2). So, \( BA \) is not defined.
2. **Dimensions of \( A \) and \( C \):**
- Matrix \( A \): 2 rows × 3 columns.
- Matrix \( C \): 3 rows × 3 columns.
- Multiplying \( A \) and \( C \) is defined and results in a matrix with dimensions 2 × 3.
Given that \( BA \) is not defined:
\[
\text{The matrix is not defined.}
\]
### Conclusion:
For the given matrices \( A \), \( B \), and \( C \), the expression \( (3)BA + (4)AC \) cannot be computed because \( BA \) is not a defined operation based on the matrix dimensions provided.
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