Math Problem Statement
If C = AB, what is C(1,2)? Given matrices A = [[1, -2, 3], [4, 0, 7]] and B = [[2, 4], [-1, 3], [0, 2]].
Solution
We are tasked with calculating the element , where . Here, and are matrices, and refers to the element in the first row and second column of matrix .
Given:
Matrix :
1 & -2 & 3 \\ 4 & 0 & 7 \end{pmatrix}$$ Matrix $$B$$: $$B = \begin{pmatrix} 2 & 4 \\ -1 & 3 \\ 0 & 2 \end{pmatrix}$$ We need to compute the matrix product $$C = AB$$, and then extract the element at position $$C(1,2)$$. ### Step 1: Matrix multiplication To multiply $$A$$ and $$B$$, the element at position $$C(i,j)$$ is given by the dot product of the $$i$$-th row of matrix $$A$$ and the $$j$$-th column of matrix $$B$$. #### Compute $$C(1,2)$$: The first row of matrix $$A$$ is: $$(1, -2, 3)$$ The second column of matrix $$B$$ is: $$(4, 3, 2)$$ Now, calculate the dot product: $$C(1,2) = (1 \times 4) + (-2 \times 3) + (3 \times 2)$$ $$C(1,2) = 4 + (-6) + 6 = 4$$ ### Final Answer: The value of $$C(1,2)$$ is **4**. --- Would you like any further explanation or details? Here are some related questions: 1. How is the entire matrix $$C$$ formed after multiplying $$A$$ and $$B$$? 2. What are the properties of matrix multiplication (e.g., commutativity)? 3. How do we determine the size of the resulting matrix in matrix multiplication? 4. What are the steps to perform matrix multiplication in larger matrices? 5. Can matrix multiplication be applied to transformations in 3D space? **Tip**: Always verify that the dimensions of matrices allow multiplication; the number of columns in the first matrix must equal the number of rows in the second.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Linear Algebra
Formulas
Matrix multiplication formula: C(i,j) = sum(A(i,k) * B(k,j))
Theorems
Matrix multiplication theorem
Suitable Grade Level
Grades 10-12