Math Problem Statement
Verify that (AB)^T = B^T A^T, where A = [[2, 5, 6], [0, 1, 2]] and B = [[6, 0, 4], [5, 7, 0]].
Solution
The problem requires verification of the matrix identity , where matrices and are given as:
\quad \text{and} \quad B = \begin{bmatrix} 6 & 0 & 4 \\ 5 & 7 & 0 \end{bmatrix}$$ ### Step 1: Compute $$AB$$ First, we calculate the product $$AB$$: $$AB = \begin{bmatrix} 2 & 5 & 6 \\ 0 & 1 & 2 \end{bmatrix} \times \begin{bmatrix} 6 & 0 & 4 \\ 5 & 7 & 0 \end{bmatrix}$$ This multiplication can be done by taking the dot product of rows from $$A$$ with columns from $$B$$: $$AB = \begin{bmatrix} (2 \times 6 + 5 \times 5) & (2 \times 0 + 5 \times 7) & (2 \times 4 + 5 \times 0) \\ (0 \times 6 + 1 \times 5) & (0 \times 0 + 1 \times 7) & (0 \times 4 + 1 \times 0) \end{bmatrix}$$ $$AB = \begin{bmatrix} 47 & 35 & 8 \\ 5 & 7 & 0 \end{bmatrix}$$ ### Step 2: Compute $$(AB)^T$$ Now, compute the transpose of the product $$AB$$: $$(AB)^T = \begin{bmatrix} 47 & 35 & 8 \\ 5 & 7 & 0 \end{bmatrix}^T = \begin{bmatrix} 47 & 5 \\ 35 & 7 \\ 8 & 0 \end{bmatrix}$$ ### Step 3: Compute $$B^T$$ and $$A^T$$ Next, we calculate the transposes of $$B$$ and $$A$$: $$B^T = \begin{bmatrix} 6 & 5 \\ 0 & 7 \\ 4 & 0 \end{bmatrix}, \quad A^T = \begin{bmatrix} 2 & 0 \\ 5 & 1 \\ 6 & 2 \end{bmatrix}$$ ### Step 4: Compute $$B^T A^T$$ Now, calculate the product $$B^T A^T$$: $$B^T A^T = \begin{bmatrix} 6 & 5 \\ 0 & 7 \\ 4 & 0 \end{bmatrix} \times \begin{bmatrix} 2 & 0 \\ 5 & 1 \\ 6 & 2 \end{bmatrix}$$ $$B^T A^T = \begin{bmatrix} (6 \times 2 + 5 \times 5) & (6 \times 0 + 5 \times 1) \\ (0 \times 2 + 7 \times 5) & (0 \times 0 + 7 \times 1) \\ (4 \times 2 + 0 \times 5) & (4 \times 0 + 0 \times 1) \end{bmatrix}$$ $$B^T A^T = \begin{bmatrix} 47 & 5 \\ 35 & 7 \\ 8 & 0 \end{bmatrix}$$ ### Step 5: Conclusion Since $$(AB)^T = B^T A^T$$, the given matrix identity is verified. Would you like more details or have any further questions? --- Here are five related questions for further exploration: 1. What are the conditions under which $$(AB)^T = B^T A^T$$ always holds? 2. How does the transpose of a matrix product differ from the product of transposes? 3. Can this property of transposes be extended to three or more matrices? 4. How does the inverse of a product of matrices relate to their individual inverses? 5. How does matrix multiplication relate to transformations in geometry? **Tip:** When multiplying matrices, make sure the dimensions are compatible—i.e., the number of columns in the first matrix must match 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 Transpose
Formulas
(AB)^T = B^T A^T
Matrix multiplication formula: (A * B)[i,j] = sum(A[i,k] * B[k,j])
Theorems
Transpose of a Product of Matrices
Suitable Grade Level
Grades 10-12