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.