give example of matrix multiplication of 2*2 and2*1 matrix

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

### Matrices:

\[
A = \begin{pmatrix} 
a_{11} & a_{12} \\
a_{21} & a_{22} 
\end{pmatrix} 
= \begin{pmatrix} 
1 & 2 \\
3 & 4 
\end{pmatrix}
\]

\[
B = \begin{pmatrix} 
b_{11} \\
b_{21} 
\end{pmatrix} 
= \begin{pmatrix} 
5 \\
6 
\end{pmatrix}
\]

### 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 \( B \times A \).

Explore an example of matrix multiplication between a 2x2 matrix and a 2x1 matrix. Learn how to perform matrix multiplication step-by-step with detailed calculations. Suitable for high school students studying matrix algebra.

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