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We are given a matrix equation, where the product of two matrices is equal to another matrix:

\[
\begin{bmatrix} 
1 & 2 \\
3 & 4 
\end{bmatrix}
\begin{bmatrix} 
3 & 1 \\
5 & 2 
\end{bmatrix}
= 
\begin{bmatrix} 
7 & 11 \\
k & 23 
\end{bmatrix}
\]

We are tasked with finding the value of \(k\).

### Step-by-Step Matrix Multiplication:

Let’s denote the two matrices we are multiplying as \( A \) and \( B \), and the result matrix as \( C \), such that:
\[
C = A \times B
\]

The multiplication of two 2x2 matrices follows this rule:
\[
C = \begin{bmatrix} 
a & b \\
c & d 
\end{bmatrix}
\begin{bmatrix} 
e & f \\
g & h 
\end{bmatrix}
=
\begin{bmatrix} 
(ae + bg) & (af + bh) \\
(ce + dg) & (cf + dh) 
\end{bmatrix}
\]

For the given matrices, we have:

\[
A = \begin{bmatrix} 
1 & 2 \\
3 & 4 
\end{bmatrix}, 
\quad 
B = \begin{bmatrix} 
3 & 1 \\
5 & 2 
\end{bmatrix}
\]

Performing the matrix multiplication:

\[
C = \begin{bmatrix} 
(1 \cdot 3 + 2 \cdot 5) & (1 \cdot 1 + 2 \cdot 2) \\
(3 \cdot 3 + 4 \cdot 5) & (3 \cdot 1 + 4 \cdot 2) 
\end{bmatrix}
= \begin{bmatrix} 
(3 + 10) & (1 + 4) \\
(9 + 20) & (3 + 8) 
\end{bmatrix}
= \begin{bmatrix} 
13 & 5 \\
29 & 11 
\end{bmatrix}
\]

We know the result matrix should be:
\[
\begin{bmatrix} 
7 & 11 \\
k & 23 
\end{bmatrix}
\]

By comparing the corresponding elements of the two matrices, we have:
\[
k = 29
\]

Therefore, the correct value of \( k \) is **29**, but this does not match any of the given answer choices. There might be an error in the question or answer choices, as the calculated value of \( k \) is correct based on the matrix multiplication. Would you like to discuss this further or explore a different approach?

Here are 5 related questions:
1. What are the rules for matrix multiplication for general \( m \times n \) and \( n \times p \) matrices?
2. Can we perform matrix multiplication on non-square matrices?
3. How do you find the inverse of a 2x2 matrix?
4. What is the determinant of a 2x2 matrix and how is it related to matrix inversion?
5. How would the value of \( k \) change if the second matrix were altered?

**Tip:** When multiplying matrices, always ensure that the number of columns in the first matrix equals the number of rows in the second matrix.

If \[ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix} = \begin{bmatrix} 7 & 11 \\ k & 23 \end{bmatrix}, then find the value of k. (A) 17 (B) 19 (C) 21 (D) 23

Learn how to find the unknown value 'k' by solving a matrix multiplication problem involving 2x2 matrices. This problem demonstrates key linear algebra concepts with step-by-step matrix multiplication, suitable for advanced high school or college-level students.

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