Math Problem Statement

A = [1, 4, 3; 6, 5, 10], B = [5, 10; 7, 2; 8, 8]. Compute for c11 in A × B.

Solution

We are given two matrices AA and BB, and are asked to compute the element c11c_{11} of their product A×BA \times B.

Matrix AA (2x3):

1 & 4 & 3 \\ 6 & 5 & 10 \end{bmatrix}$$ ### Matrix $$B$$ (3x2): $$B = \begin{bmatrix} 5 & 10 \\ 7 & 2 \\ 8 & 8 \end{bmatrix}$$ To compute $$c_{11}$$ (the element in the first row and first column of the product matrix $$C = A \times B$$), we perform the dot product of the first row of $$A$$ and the first column of $$B$$. $$c_{11} = (1 \times 5) + (4 \times 7) + (3 \times 8)$$ Now, I'll calculate that.The value of $$c_{11}$$ is $$57$$. Would you like further details or explanations? Here are 5 related questions to expand your understanding: 1. How would you calculate $$c_{12}$$ for the same matrix multiplication? 2. What is the process to find all elements of the product matrix $$C = A \times B$$? 3. How can you check if two matrices are compatible for multiplication? 4. What is the geometrical interpretation of matrix multiplication? 5. Can you perform matrix multiplication with non-square matrices? **Tip**: In matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix for the product to be defined.

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Math Problem Analysis

Mathematical Concepts

Matrix Multiplication
Linear Algebra

Formulas

Matrix multiplication formula: c_{ij} = Σ (A_{ik} * B_{kj})

Theorems

Properties of Matrix Multiplication

Suitable Grade Level

Grades 10-12