Math Problem Statement

Given the following matrices, determine 4A - 3B if possible. If not, state 'Not Possible'. A = [[5, 8, -7], [0, 0, -9]] and B = [[-8, 8, -8], [-5, 9, 0]]

Solution

The problem asks you to determine the result of the matrix operation 4A3B4A - 3B, or state "Not Possible" if the operation cannot be performed. Let's go through it step by step.


Given Matrices:

\begin{bmatrix} 5 & 8 & -7 \\ 0 & 0 & -9 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} -8 & 8 & -8 \\ -5 & 9 & 0 \end{bmatrix}$$ --- ### Step 1: Verify Matrix Dimensions - $$A$$ is a $$2 \times 3$$ matrix. - $$B$$ is also a $$2 \times 3$$ matrix. Since both matrices have the same dimensions, we can proceed with the matrix subtraction. --- ### Step 2: Perform the Scalar Multiplications #### Multiply $$A$$ by 4: $$4A = 4 \cdot \begin{bmatrix} 5 & 8 & -7 \\ 0 & 0 & -9 \end{bmatrix} = \begin{bmatrix} 20 & 32 & -28 \\ 0 & 0 & -36 \end{bmatrix}$$ #### Multiply $$B$$ by 3: $$3B = 3 \cdot \begin{bmatrix} -8 & 8 & -8 \\ -5 & 9 & 0 \end{bmatrix} = \begin{bmatrix} -24 & 24 & -24 \\ -15 & 27 & 0 \end{bmatrix}$$ --- ### Step 3: Subtract $$3B$$ from $$4A$$ $$4A - 3B = \begin{bmatrix} 20 & 32 & -28 \\ 0 & 0 & -36 \end{bmatrix} - \begin{bmatrix} -24 & 24 & -24 \\ -15 & 27 & 0 \end{bmatrix}$$ Now, subtract the corresponding elements: $$4A - 3B = \begin{bmatrix} 20 - (-24) & 32 - 24 & -28 - (-24) \\ 0 - (-15) & 0 - 27 & -36 - 0 \end{bmatrix}$$ $$4A - 3B = \begin{bmatrix} 44 & 8 & -4 \\ 15 & -27 & -36 \end{bmatrix}$$ --- ### Final Answer: $$4A - 3B = \begin{bmatrix} 44 & 8 & -4 \\ 15 & -27 & -36 \end{bmatrix}$$ --- Let me know if you need further clarification or additional steps! --- ### Relative Questions: 1. How do you determine if two matrices can be added or subtracted? 2. What are the rules for scalar multiplication of matrices? 3. How can matrix operations be applied in real-world problems? 4. What happens if the matrices have different dimensions? 5. Can you explain how element-wise operations work in matrices? --- ### Tip: When performing matrix operations, always ensure the matrices have compatible dimensions to avoid mistakes.

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Matrix Subtraction
Scalar Multiplication

Formulas

Matrix scalar multiplication: k * A
Matrix subtraction: A - B

Theorems

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Suitable Grade Level

High School (Grades 10-12)