Math Problem Statement
Find A x B for the given matrices:
A =
[[-4, 7],
[3, -8],
[1, 6]]
B =
[[8, 7, -1],
[-3, 5, 4]]
Solution
The problem requires us to calculate the matrix product , where the given matrices are:
\begin{bmatrix} -4 & 7 \\ 3 & -8 \\ 1 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 8 & 7 & -1 \\ -3 & 5 & 4 \end{bmatrix}.$$ ### Steps for Matrix Multiplication Matrix multiplication follows the rule: \[ C[i][j] = \sum_{k=1}^n A[i][k] \cdot B[k][j], \] where $$C[i][j]$$ is the element of the resultant matrix $$C$$ at row $$i$$ and column $$j$$. Here: - $$A$$ is a $$3 \times 2$$ matrix. - $$B$$ is a $$2 \times 3$$ matrix. - The result $$C$$ will be a $$3 \times 3$$ matrix. --- ### Solution: Let's calculate $$C = A \times B$$: #### Row 1 of $$A$$ and Columns of $$B$$: \[ C[1][1] = (-4)(8) + (7)(-3) = -32 - 21 = -53, \] \[ C[1][2] = (-4)(7) + (7)(5) = -28 + 35 = 7, \] \[ C[1][3] = (-4)(-1) + (7)(4) = 4 + 28 = 32. \] #### Row 2 of $$A$$ and Columns of $$B$$: \[ C[2][1] = (3)(8) + (-8)(-3) = 24 + 24 = 48, \] \[ C[2][2] = (3)(7) + (-8)(5) = 21 - 40 = -19, \] \[ C[2][3] = (3)(-1) + (-8)(4) = -3 - 32 = -35. \] #### Row 3 of $$A$$ and Columns of $$B$$: \[ C[3][1] = (1)(8) + (6)(-3) = 8 - 18 = -10, \] \[ C[3][2] = (1)(7) + (6)(5) = 7 + 30 = 37, \] \[ C[3][3] = (1)(-1) + (6)(4) = -1 + 24 = 23. \] --- ### Final Result: $$C = \begin{bmatrix} -53 & 7 & 32 \\ 48 & -19 & -35 \\ -10 & 37 & 23 \end{bmatrix}.$$ Let me know if you'd like a step-by-step breakdown of any part of the computation! --- ### Follow-Up Questions: 1. What is the general rule for determining the dimensions of a resulting matrix from a product $$A \times B$$? 2. How would the multiplication change if one of the matrices were a square matrix (e.g., $$2 \times 2$$)? 3. Can a $$3 \times 2$$ matrix multiply with a $$3 \times 3$$ matrix? Why or why not? 4. What are some real-world applications of matrix multiplication? 5. How does matrix multiplication differ from element-wise multiplication? --- **Tip:** Always ensure the inner dimensions of matrices align 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
C[i][j] = Σ (A[i][k] × B[k][j]) for k = 1 to n
Theorems
-
Suitable Grade Level
Grades 10-12 or Undergraduate