Math Problem Statement
Given A = [[0, 3, 4], [4, -1, 2]], B = [[7, 9, -1], [6, -2, 3]], C = [[12, 6, 4], [9, 1, 0]], perform the operation: 2(A - B) + 4(C - A) + 2(A - C) - 2C + 3B.
Solution
The problem given asks to perform the following matrix operation:
where the matrices are defined as:
We will break this down step-by-step.
Step 1: Compute
= \begin{bmatrix} -7 & -6 & 5 \\ -2 & 1 & -1 \end{bmatrix}$$ ### Step 2: Compute $$C - A$$ $$C - A = \begin{bmatrix} 12 & 6 & 4 \\ 9 & 1 & 0 \end{bmatrix} - \begin{bmatrix} 0 & 3 & 4 \\ 4 & -1 & 2 \end{bmatrix} = \begin{bmatrix} 12 - 0 & 6 - 3 & 4 - 4 \\ 9 - 4 & 1 + 1 & 0 - 2 \end{bmatrix} = \begin{bmatrix} 12 & 3 & 0 \\ 5 & 2 & -2 \end{bmatrix}$$ ### Step 3: Compute $$A - C$$ $$A - C = \begin{bmatrix} 0 & 3 & 4 \\ 4 & -1 & 2 \end{bmatrix} - \begin{bmatrix} 12 & 6 & 4 \\ 9 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 - 12 & 3 - 6 & 4 - 4 \\ 4 - 9 & -1 - 1 & 2 - 0 \end{bmatrix} = \begin{bmatrix} -12 & -3 & 0 \\ -5 & -2 & 2 \end{bmatrix}$$ ### Step 4: Multiply and sum terms Now we apply the multiplications and sums from the original expression: 1. $$2(A - B) = 2 \times \begin{bmatrix} -7 & -6 & 5 \\ -2 & 1 & -1 \end{bmatrix} = \begin{bmatrix} -14 & -12 & 10 \\ -4 & 2 & -2 \end{bmatrix}$$ 2. $$4(C - A) = 4 \times \begin{bmatrix} 12 & 3 & 0 \\ 5 & 2 & -2 \end{bmatrix} = \begin{bmatrix} 48 & 12 & 0 \\ 20 & 8 & -8 \end{bmatrix}$$ 3. $$2(A - C) = 2 \times \begin{bmatrix} -12 & -3 & 0 \\ -5 & -2 & 2 \end{bmatrix} = \begin{bmatrix} -24 & -6 & 0 \\ -10 & -4 & 4 \end{bmatrix}$$ 4. $$-2C = -2 \times \begin{bmatrix} 12 & 6 & 4 \\ 9 & 1 & 0 \end{bmatrix} = \begin{bmatrix} -24 & -12 & -8 \\ -18 & -2 & 0 \end{bmatrix}$$ 5. $$3B = 3 \times \begin{bmatrix} 7 & 9 & -1 \\ 6 & -2 & 3 \end{bmatrix} = \begin{bmatrix} 21 & 27 & -3 \\ 18 & -6 & 9 \end{bmatrix}$$ ### Step 5: Combine all terms Now we add all the resulting matrices: $$2(A - B) + 4(C - A) + 2(A - C) - 2C + 3B = \begin{bmatrix} -14 & -12 & 10 \\ -4 & 2 & -2 \end{bmatrix} + \begin{bmatrix} 48 & 12 & 0 \\ 20 & 8 & -8 \end{bmatrix} + \begin{bmatrix} -24 & -6 & 0 \\ -10 & -4 & 4 \end{bmatrix} + \begin{bmatrix} -24 & -12 & -8 \\ -18 & -2 & 0 \end{bmatrix} + \begin{bmatrix} 21 & 27 & -3 \\ 18 & -6 & 9 \end{bmatrix}$$ Now adding the elements together: $$= \begin{bmatrix} (-14 + 48 - 24 - 24 + 21) & (-12 + 12 - 6 - 12 + 27) & (10 + 0 + 0 - 8 - 3) \\ (-4 + 20 - 10 - 18 + 18) & (2 + 8 - 4 - 2 - 6) & (-2 - 8 + 4 + 0 + 9) \end{bmatrix}$$ $$= \begin{bmatrix} 7 & 9 & -1 \\ 6 & -2 & 3 \end{bmatrix}$$ Thus, the final result of the operation is: $$\boxed{\begin{bmatrix} 7 & 9 & -1 \\ 6 & -2 & 3 \end{bmatrix}}$$ This matches the first answer option. Would you like further details or clarifications? ### Follow-up Questions: 1. What are the conditions for matrix addition or subtraction to be valid? 2. How do you determine if two matrices can be multiplied? 3. What are the steps for multiplying a matrix by a scalar? 4. How can you verify the correctness of matrix operations? 5. What are some common properties of matrix operations (e.g., distributive property)? ### Tip: Always ensure the dimensions of the matrices match for addition or subtraction operations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
Matrix Addition
Matrix Subtraction
Scalar Multiplication
Formulas
Matrix addition: (A + B) = (a_ij + b_ij)
Matrix subtraction: (A - B) = (a_ij - b_ij)
Scalar multiplication: cA = c * a_ij
Theorems
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Suitable Grade Level
University level (Linear Algebra)