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We are given two matrices \( A \) and \( B \), and are asked to compute the element \( c_{11} \) of their product \( A \times B \).

### Matrix \( A \) (2x3):
\[
A = \begin{bmatrix}
1 & 4 & 3 \\
6 & 5 & 10
\end{bmatrix}
\]

### 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.

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

Learn how to compute the element c11 from the matrix multiplication of A and B. This problem involves key concepts from linear algebra, focusing on matrix multiplication. Suitable for students in Grades 10-12.

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