Math Problem Statement
(8 5 (5 9 12 16 * 3 7) 0 13 5 0)
Matrices calculate
Solution
To multiply the given matrices, we need to follow the standard matrix multiplication procedure. Here are the matrices you provided:
\quad \text{and} \quad B = \begin{pmatrix} 5 & 9 \\ 3 & 7 \end{pmatrix}$$ ### Step-by-Step Multiplication: Matrix $$A$$ is a $$4 \times 2$$ matrix, and matrix $$B$$ is a $$2 \times 2$$ matrix. The resulting matrix $$C$$ will have dimensions $$4 \times 2$$. $$C = AB = \begin{pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \\ c_{31} & c_{32} \\ c_{41} & c_{42} \end{pmatrix}$$ Each element $$c_{ij}$$ of matrix $$C$$ is calculated as: $$c_{ij} = \sum_{k=1}^{2} a_{ik} \cdot b_{kj}$$ **Calculating each element:** 1. **$$c_{11}$$:** $$c_{11} = (8 \times 5) + (5 \times 3) = 40 + 15 = 55$$ 2. **$$c_{12}$$:** $$c_{12} = (8 \times 9) + (5 \times 7) = 72 + 35 = 107$$ 3. **$$c_{21}$$:** $$c_{21} = (12 \times 5) + (16 \times 3) = 60 + 48 = 108$$ 4. **$$c_{22}$$:** $$c_{22} = (12 \times 9) + (16 \times 7) = 108 + 112 = 220$$ 5. **$$c_{31}$$:** $$c_{31} = (0 \times 5) + (13 \times 3) = 0 + 39 = 39$$ 6. **$$c_{32}$$:** $$c_{32} = (0 \times 9) + (13 \times 7) = 0 + 91 = 91$$ 7. **$$c_{41}$$:** $$c_{41} = (5 \times 5) + (0 \times 3) = 25 + 0 = 25$$ 8. **$$c_{42}$$:** $$c_{42} = (5 \times 9) + (0 \times 7) = 45 + 0 = 45$$ ### Final Matrix $$C$$: $$C = \begin{pmatrix} 55 & 107 \\ 108 & 220 \\ 39 & 91 \\ 25 & 45 \end{pmatrix}$$ Would you like further details on any of these steps? --- ### Related Questions: 1. What are the rules for matrix multiplication? 2. Can matrix multiplication be commutative? 3. How do you find the determinant of a matrix? 4. What is an identity matrix and its role in multiplication? 5. How do you compute the inverse of a matrix? 6. What are the applications of matrix multiplication in real life? 7. Can two matrices of different dimensions be multiplied? 8. How does matrix multiplication differ from element-wise multiplication? ### Tip: When multiplying matrices, 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
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Formulas
Matrix multiplication formula
Theorems
-
Suitable Grade Level
High School