Math Problem Statement
This exercise should be done two ways: by hand and using technology where possible. Let A =
2 −2
0 6 0 −6 , B =
3 0 −1
4 −1 1 , C =
x 1 w
z r 4 . Evaluate the following. A(B + C)
Solution
To evaluate , we first compute the matrix , then multiply it by matrix .
Let’s break this down step by step:
Step 1: Write the matrices
Given:
B = \begin{bmatrix} 3 & 0 & -1 \\ 4 & -1 & 1 \end{bmatrix}, \quad C = \begin{bmatrix} x & 1 & w \\ z & r & 4 \end{bmatrix}.$$ ### Step 2: Add matrices $$B$$ and $$C$$ $$B + C = \begin{bmatrix} 3 & 0 & -1 \\ 4 & -1 & 1 \end{bmatrix} + \begin{bmatrix} x & 1 & w \\ z & r & 4 \end{bmatrix} = \begin{bmatrix} 3+x & 1 & -1+w \\ 4+z & -1+r & 1+4 \end{bmatrix}.$$ So: $$B + C = \begin{bmatrix} 3+x & 1 & -1+w \\ 4+z & -1+r & 5 \end{bmatrix}.$$ ### Step 3: Multiply $$A$$ by $$B + C$$ Now we compute $$A(B + C) = A \cdot (B + C)$$: $$A \cdot (B + C) = \begin{bmatrix} 2 & -2 \\ 0 & 6 \\ 0 & -6 \end{bmatrix} \cdot \begin{bmatrix} 3+x & 1 & -1+w \\ 4+z & -1+r & 5 \end{bmatrix}.$$ #### Multiply row 1 of $$A$$ by $$B + C$$: $$\text{Row 1 of Result} = [2(3+x) + (-2)(4+z), 2(1) + (-2)(-1+r), 2(-1+w) + (-2)(5)].$$ Simplify: $$\text{Row 1 of Result} = [6 + 2x - 8 - 2z, 2 + 2r, -2 + 2w - 10].$$ $$\text{Row 1 of Result} = [-2 + 2x - 2z, 2 + 2r, -12 + 2w].$$ #### Multiply row 2 of $$A$$ by $$B + C$$: $$\text{Row 2 of Result} = [0(3+x) + 6(4+z), 0(1) + 6(-1+r), 0(-1+w) + 6(5)].$$ Simplify: $$\text{Row 2 of Result} = [24 + 6z, -6 + 6r, 30].$$ #### Multiply row 3 of $$A$$ by $$B + C$$: $$\text{Row 3 of Result} = [0(3+x) + (-6)(4+z), 0(1) + (-6)(-1+r), 0(-1+w) + (-6)(5)].$$ Simplify: $$\text{Row 3 of Result} = [-24 - 6z, 6 - 6r, -30].$$ ### Step 4: Combine the rows into the result matrix The result matrix is: $$A(B + C) = \begin{bmatrix} -2 + 2x - 2z & 2 + 2r & -12 + 2w \\ 24 + 6z & -6 + 6r & 30 \\ -24 - 6z & 6 - 6r & -30 \end{bmatrix}.$$ --- If you'd like further details on interpreting this or solving a specific case (e.g., with specific values for $$x, w, z, r$$), let me know! Would you like me to expand this explanation further? ### Related Questions: 1. What happens to the result if $$x, w, z, r$$ are specific numerical values? 2. How do matrix dimensions affect whether $$AB$$ is valid? 3. What changes if $$C$$ is transposed before the addition $$B + C$$? 4. Can this operation be performed using a computational tool (e.g., Python, MATLAB)? 5. How can we verify the distributive property $$A(B+C) = AB + AC$$? ### Tip: Always verify matrix dimensions before performing multiplication; mismatched dimensions result in invalid operations!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Operations
Matrix Addition
Matrix Multiplication
Formulas
Matrix addition: (A + B)ij = Aij + Bij
Matrix multiplication: (AB)ij = Σ(Aik * Bkj)
Theorems
Distributive Property of Matrices: A(B + C) = AB + AC
Suitable Grade Level
Grades 11-12 or College Level
Related Recommendation
Matrix Multiplication and Addition Verification: AC, BC, and (A+B)C
Matrix Operations for A, B, and C: Calculating 2A - C and A + BC
Matrix Calculation for 2A + B + 3C with Given Matrices A, B, and C
Matrix Operations and Verification of Properties: AB, AC, and Distributive Rules
Matrix Operations: 2A + BC Calculation