Math Problem Statement
If A = [[2, 1, -2], [3, 2, -4], [-4, -2, 4]] and B = [[-2, 3, 4], [2, 0, -2], [-2, 2, 1]], then determine whether AB = BA for these two square matrices.
Solution
It looks like you're asking about a matrix problem involving the multiplication of two matrices and . To solve this, we need to first compute the matrix products and , and then determine if .
Let's begin by recalling the two matrices:
2 & 1 & -2 \\ 3 & 2 & -4 \\ -4 & -2 & 4 \end{pmatrix}$$ $$B = \begin{pmatrix} -2 & 3 & 4 \\ 2 & 0 & -2 \\ -2 & 2 & 1 \end{pmatrix}$$ ### Step 1: Compute $$AB$$ To multiply matrices, we perform the dot product of the rows of $$A$$ with the columns of $$B$$. ### Step 2: Compute $$BA$$ Similarly, to compute $$BA$$, we take the dot product of the rows of $$B$$ with the columns of $$A$$. Let me calculate these products for you. numpy as np # Define matrices A and B A = np.array([[2, 1, -2], [3, 2, -4], [-4, -2, 4]]) B = np.array([[-2, 3, 4], [2, 0, -2], [-2, 2, 1]]) # Compute AB and BA AB = np.dot(A, B) BA = np.dot(B, A) AB, BAThe results of the matrix multiplications are as follows: $$AB = \begin{pmatrix} 2 & 2 & 4 \\ 6 & 1 & 4 \\ -4 & -4 & -8 \end{pmatrix}$$ $$BA = \begin{pmatrix} -11 & -4 & 8 \\ 12 & 6 & -12 \\ -2 & 0 & 0 \end{pmatrix}$$ As you can see, $$AB \neq BA$$, which means that matrix multiplication is not commutative for these two matrices. Therefore, the statement $$AB = BA$$ is **False**. Would you like any further details or clarifications on this? Here are a few related questions you might find useful: 1. What does it mean for matrix multiplication to be commutative? 2. How do we perform matrix multiplication step by step? 3. What are some other properties of matrix multiplication? 4. Can non-square matrices have commutative multiplication? 5. What happens if matrices have different dimensions for multiplication? Tip: When dealing with matrix multiplication, always ensure that the number of columns in the first matrix matches the number of rows in the second matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Matrix Commutativity
Formulas
AB = A * B
BA = B * A
Theorems
Matrix multiplication is not commutative in general
Suitable Grade Level
Grades 9-12