Example of Matrix Multiplication: 2x2 and 2x1 Matrices

Math Problem Statement

give example of matrix multiplication of 22 and21 matrix

To illustrate matrix multiplication between a $2 \times 2$ matrix and a $2 \times 1$ matrix, let's take the following example:

a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$ $$B = \begin{pmatrix} b_{11} \\ b_{21} \end{pmatrix} = \begin{pmatrix} 5 \\ 6 \end{pmatrix}$$ ### Multiplication Process: The product $$C = A \times B$$ is calculated as follows: $$C = \begin{pmatrix} (a_{11} \times b_{11}) + (a_{12} \times b_{21}) \\ (a_{21} \times b_{11}) + (a_{22} \times b_{21}) \end{pmatrix} = \begin{pmatrix} (1 \times 5) + (2 \times 6) \\ (3 \times 5) + (4 \times 6) \end{pmatrix}$$ ### Solution: $$C = \begin{pmatrix} 5 + 12 \\ 15 + 24 \end{pmatrix} = \begin{pmatrix} 17 \\ 39 \end{pmatrix}$$ So, the resulting matrix $$C$$ is: $$C = \begin{pmatrix} 17 \\ 39 \end{pmatrix}$$ Would you like further details or explanations on this? Here are some related questions to consider: 1. What is the general formula for matrix multiplication? 2. How do you multiply two matrices with different dimensions? 3. What are the conditions for matrix multiplication to be valid? 4. How is matrix multiplication used in solving systems of linear equations? 5. Can matrix multiplication be commutative? Why or why not? **Tip:** Matrix multiplication is not commutative; the order in which you multiply matrices matters. $$A \times B$$ is generally not the same as $$B \times A$$.

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Matrix Multiplication

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