Math Problem Statement

Find the product of the following two matrices: A = [[0, -2, 0, -1], [-1, 4, -2, 0]] and B = [[-5, 2], [1, 2], [3, -1], [4, 1]]

Solution

The problem requires calculating the product of the given two matrices:

\begin{bmatrix} 0 & -2 & 0 & -1 \\ -1 & 4 & -2 & 0 \end{bmatrix}$$ and $$\mathbf{B} = \begin{bmatrix} -5 & 2 \\ 1 & 2 \\ 3 & -1 \\ 4 & 1 \end{bmatrix}.$$ To compute the matrix product $$\mathbf{C} = \mathbf{A} \times \mathbf{B}$$, each element $$c_{ij}$$ of $$\mathbf{C}$$ is obtained as the dot product of the $$i$$-th row of $$\mathbf{A}$$ with the $$j$$-th column of $$\mathbf{B}$$. Let me compute this step by step. numpy as np # Define matrices A and B A = np.array([[0, -2, 0, -1], [-1, 4, -2, 0]]) B = np.array([[-5, 2], [1, 2], [3, -1], [4, 1]]) # Calculate the product C = A * B C = np.dot(A, B) CThe product of the two matrices is: $$\mathbf{C} = \begin{bmatrix} -6 & -5 \\ 3 & 8 \end{bmatrix}.$$ Would you like a detailed step-by-step breakdown of the computation? ### Related Questions: 1. How do you verify the dimensions of matrices for multiplication? 2. Can the order of multiplication change the result of two matrices? 3. What are some practical applications of matrix multiplication in real-world problems? 4. How does the dot product concept apply to the multiplication of matrices? 5. What happens if one or both matrices contain all zeros? ### Tip: Always verify the dimensions of the matrices before attempting multiplication: for $$A \times B$$, the number of columns in $$A$$ must equal the number of rows in $$B$$.

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Math Problem Analysis

Mathematical Concepts

Matrix Multiplication
Linear Algebra

Formulas

C = A × B, where c_ij = Σ(a_ik * b_kj)

Theorems

Matrix Dot Product

Suitable Grade Level

Grades 11-12