Math Problem Statement
For the following exercises, perform the requested operations on the given matrices. A=[41−23],B=[6117−2−34],C=⎡⎣⎢611147−20⎤⎦⎥,D=⎡⎣⎢1102−4589−75⎤⎦⎥,E=⎡⎣⎢720−14−11339⎤⎦⎥
ED
Solution
To compute the product , let's calculate it step by step. First, ensure the dimensions are compatible for matrix multiplication.
Matrices:
7 & 20 & -1 \\ 4 & -1 & -11 \\ 3 & 3 & 9 \end{bmatrix}, \quad D = \begin{bmatrix} 1 & 1 & 0 \\ 2 & -4 & 5 \\ 8 & 9 & -7 \end{bmatrix}.$$ ### Dimensions: - $$E$$ is a $$3 \times 3$$ matrix. - $$D$$ is also a $$3 \times 3$$ matrix. Since the inner dimensions match, $$E \cdot D$$ is defined, and the result will be a $$3 \times 3$$ matrix. --- ### Formula: The product of two matrices $$E = [e_{ij}]$$ and $$D = [d_{ij}]$$ is given by: $$(E \cdot D)_{ij} = \sum_{k=1}^{3} e_{ik} \cdot d_{kj}.$$ --- ### Calculation: #### First row of $$E$$ multiplied by all columns of $$D$$: 1. $$(ED)_{11} = 7(1) + 20(2) + (-1)(8) = 7 + 40 - 8 = 39$$, 2. $$(ED)_{12} = 7(1) + 20(-4) + (-1)(9) = 7 - 80 - 9 = -82$$, 3. $$(ED)_{13} = 7(0) + 20(5) + (-1)(-7) = 0 + 100 + 7 = 107$$. #### Second row of $$E$$ multiplied by all columns of $$D$$: 4. $$(ED)_{21} = 4(1) + (-1)(2) + (-11)(8) = 4 - 2 - 88 = -86$$, 5. $$(ED)_{22} = 4(1) + (-1)(-4) + (-11)(9) = 4 + 4 - 99 = -91$$, 6. $$(ED)_{23} = 4(0) + (-1)(5) + (-11)(-7) = 0 - 5 + 77 = 72$$. #### Third row of $$E$$ multiplied by all columns of $$D$$: 7. $$(ED)_{31} = 3(1) + 3(2) + 9(8) = 3 + 6 + 72 = 81$$, 8. $$(ED)_{32} = 3(1) + 3(-4) + 9(9) = 3 - 12 + 81 = 72$$, 9. $$(ED)_{33} = 3(0) + 3(5) + 9(-7) = 0 + 15 - 63 = -48$$. --- ### Result: The matrix product $$E \cdot D$$ is: $$E \cdot D = \begin{bmatrix} 39 & -82 & 107 \\ -86 & -91 & 72 \\ 81 & 72 & -48 \end{bmatrix}.$$ Let me know if you'd like further clarifications or additional examples! --- ### 5 Related Questions: 1. How do you determine if two matrices are compatible for multiplication? 2. Can you compute $$D \cdot E$$ for these same matrices? 3. What properties of matrix multiplication make it non-commutative? 4. How does matrix multiplication relate to linear transformations? 5. Can the determinant of $$E \cdot D$$ be directly derived from $$E$$ and $$D$$? --- ### Tip: Always verify the dimensions of the matrices before attempting to multiply them!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Linear Algebra
Formulas
Matrix product formula: (AB)_{ij} = Σ(A_{ik} * B_{kj})
Theorems
Associativity of Matrix Multiplication
Dimension Compatibility Theorem
Suitable Grade Level
Grades 10-12 or early undergraduate level