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The problem asks to determine if the matrix multiplication \( AB \) is possible for the given matrices \( A \) and \( B \), and if it is, to find the resulting matrix and its dimensions.

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

### Step 1: Check Matrix Dimensions

- Matrix \( A \) is a column vector with 3 rows and 1 column, i.e., \( A \) is of size \( 3 \times 1 \).
- Matrix \( B \) is a row vector with 1 row and 3 columns, i.e., \( B \) is of size \( 1 \times 3 \).

### Step 2: Conditions for Matrix Multiplication

Matrix multiplication \( AB \) is possible if the number of columns in \( A \) is equal to the number of rows in \( B \). However:

- Matrix \( A \) has 1 column.
- Matrix \( B \) has 1 row.

Since the number of columns in \( A \) (which is 1) matches the number of rows in \( B \) (which is also 1), **matrix multiplication is possible**.

### Step 3: Resulting Matrix Dimensions

When multiplying an \( m \times n \) matrix by an \( n \times p \) matrix, the resulting matrix will have dimensions \( m \times p \).

- \( A \) is \( 3 \times 1 \).
- \( B \) is \( 1 \times 3 \).

Thus, the resulting matrix \( AB \) will have dimensions \( 3 \times 3 \).

### Step 4: Matrix Multiplication

We can now compute the matrix multiplication \( AB \):

\[
AB = \begin{bmatrix} -4 \\ 9 \\ -2 \end{bmatrix} \times \begin{bmatrix} 6 & 3 & -6 \end{bmatrix} = \begin{bmatrix} -4 \times 6 & -4 \times 3 & -4 \times (-6) \\ 9 \times 6 & 9 \times 3 & 9 \times (-6) \\ -2 \times 6 & -2 \times 3 & -2 \times (-6) \end{bmatrix}
\]

Calculating each element:

\[
AB = \begin{bmatrix} -24 & -12 & 24 \\ 54 & 27 & -54 \\ -12 & -6 & 12 \end{bmatrix}
\]

Thus, the resulting matrix is:

\[
AB = \begin{bmatrix} -24 & -12 & 24 \\ 54 & 27 & -54 \\ -12 & -6 & 12 \end{bmatrix}
\]

### Conclusion

The matrix multiplication \( AB \) is possible, and the resulting matrix is a \( 3 \times 3 \) matrix as shown above.

Would you like further explanation on any part of the solution? Here are five related questions to consider:

1. How do you determine when matrix multiplication is possible?
2. What are the properties of matrix multiplication (e.g., associative, distributive)?
3. Can you multiply \( B \times A \) in this case? Why or why not?
4. How does the rank of a matrix affect the outcome of matrix multiplication?
5. What are some real-world applications of matrix multiplication?

**Tip:** When multiplying matrices, always ensure that the number of columns in the first matrix matches the number of rows in the second matrix to make sure the operation is valid.

Given the following matrices, if possible, determine AB. Identify the dimensions of the resulting matrix and fill out the matrix if it exists. If not, state 'Not Possible'. A = [-4, 9, -2], B = [6, 3, -6]

Learn how to determine if the matrix multiplication AB is possible for the matrices A = [-4, 9, -2] and B = [6, 3, -6]. This problem explores key matrix multiplication rules and the dimensionality condition for multiplying matrices. Suitable for high school and college students.

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