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The problem asks to perform the operation `2A * 2B` using the given matrices \( A \) and \( B \). Let's break down the steps:

1. **Multiply matrix \( A \) by 2**:
   \[
   2A = 2 \times A = 2 \times \begin{bmatrix}
   -2 & 12 & 0 \\
   25 & -8 & 9
   \end{bmatrix}
   = \begin{bmatrix}
   -4 & 24 & 0 \\
   50 & -16 & 18
   \end{bmatrix}
   \]

2. **Multiply matrix \( B \) by 2**:
   \[
   2B = 2 \times B = 2 \times \begin{bmatrix}
   4 & 10 & 18 \\
   5 & -4 & 11 \\
   6 & 55 & 23
   \end{bmatrix}
   = \begin{bmatrix}
   8 & 20 & 36 \\
   10 & -8 & 22 \\
   12 & 110 & 46
   \end{bmatrix}
   \]

3. **Matrix Multiplication \( 2A * 2B \)**:
   We need to perform matrix multiplication of \( 2A \) and \( 2B \). For matrix multiplication, the number of columns in the first matrix must match the number of rows in the second matrix. In this case, \( 2A \) is a \( 2 \times 3 \) matrix, and \( 2B \) is a \( 3 \times 3 \) matrix. The resulting matrix will be a \( 2 \times 3 \) matrix.

I will now calculate \( 2A * 2B \).The result of the matrix multiplication \( 2A * 2B \) is:

\[
2A * 2B = \begin{bmatrix}
208 & -272 & 384 \\
456 & 3108 & 2276
\end{bmatrix}
\]

Would you like further details or have any questions about this operation?

Here are some related questions that might help deepen your understanding:

1. How do you determine if two matrices can be multiplied?
2. What is the significance of the dimensions of the resulting matrix in matrix multiplication?
3. Can you explain why matrix multiplication is not commutative (i.e., why \( AB \neq BA \))?
4. How can matrix multiplication be applied in practical scenarios like transformations?
5. What are some properties of matrix operations that are useful in linear algebra?

**Tip:** When performing matrix operations in programming languages like MATLAB or Python, always ensure the matrix dimensions align with the operation requirements (e.g., multiplication, addition).

For the following matrices, perform the operations in Matlab: 2A * 2B. Matrices A and B are as follows: A = [[-2, 12, 0], [25, -8, 9]], B = [[4, 10, 18], [5, -4, 11], [6, 55, 23]]

Learn how to perform matrix multiplication with scalar factors using the given matrices A and B. This tutorial includes step-by-step operations for scalar multiplication and the final matrix multiplication. Suitable for undergraduate students studying linear algebra.

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