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To find \( A \times B \), we need to perform matrix multiplication of matrices \( A \) and \( B \).

Given:
\[
A = \begin{bmatrix} 3 & 8 & 6 \\ -9 & -7 & -6 \\ 3 & 5 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 4 & 6 \\ -5 & 1 & 6 \\ -2 & 9 & 1 \end{bmatrix}
\]

Each element in the resulting matrix \( C = A \times B \) is calculated by taking the dot product of the rows of \( A \) with the columns of \( B \).

Let’s calculate each element in \( C \):

1. **First row of \( C \):**
   - \( C_{11} = (3 \cdot 2) + (8 \cdot -5) + (6 \cdot -2) = 6 - 40 - 12 = -46 \)
   - \( C_{12} = (3 \cdot 4) + (8 \cdot 1) + (6 \cdot 9) = 12 + 8 + 54 = 74 \)
   - \( C_{13} = (3 \cdot 6) + (8 \cdot 6) + (6 \cdot 1) = 18 + 48 + 6 = 72 \)

2. **Second row of \( C \):**
   - \( C_{21} = (-9 \cdot 2) + (-7 \cdot -5) + (-6 \cdot -2) = -18 + 35 + 12 = 29 \)
   - \( C_{22} = (-9 \cdot 4) + (-7 \cdot 1) + (-6 \cdot 9) = -36 - 7 - 54 = -97 \)
   - \( C_{23} = (-9 \cdot 6) + (-7 \cdot 6) + (-6 \cdot 1) = -54 - 42 - 6 = -102 \)

3. **Third row of \( C \):**
   - \( C_{31} = (3 \cdot 2) + (5 \cdot -5) + (5 \cdot -2) = 6 - 25 - 10 = -29 \)
   - \( C_{32} = (3 \cdot 4) + (5 \cdot 1) + (5 \cdot 9) = 12 + 5 + 45 = 62 \)
   - \( C_{33} = (3 \cdot 6) + (5 \cdot 6) + (5 \cdot 1) = 18 + 30 + 5 = 53 \)

Thus, the resulting matrix \( C = A \times B \) is:
\[
C = \begin{bmatrix} -46 & 74 & 72 \\ 29 & -97 & -102 \\ -29 & 62 & 53 \end{bmatrix}
\]

Would you like more details or have any questions?

### Related Questions
1. What are the rules for matrix multiplication?
2. How would the result change if matrix \( B \) were transposed?
3. Can we multiply any two matrices? Why or why not?
4. How can we interpret matrix multiplication in terms of transformations?
5. What are some real-life applications of matrix multiplication?

### Tip
For matrix multiplication, ensure the number of columns in the first matrix matches the number of rows in the second matrix.

Find the matrix multiplication of A and B, where A = [[3, 8, 6], [-9, -7, -6], [3, 5, 5]] and B = [[2, 4, 6], [-5, 1, 6], [-2, 9, 1]].

Calculate the matrix product of two 3x3 matrices A and B, with step-by-step guidance on each element of the resulting matrix. This problem covers essential linear algebra concepts, matrix multiplication rules, and calculations for Grades 10-12.

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