Question: Find the product of the following matrices, if possible.
$$\begin{bmatrix}-9 & -6 \\ -4 & 4\end{bmatrix} \begin{bmatrix}4 \\ -5\end{bmatrix}$$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
$$\begin{bmatrix}-9 & -6 \\ -4 & 4\end{bmatrix} \begin{bmatrix}4 \\ -5\end{bmatrix} = \square$$ (Simplify your answer.)
B.
The product is not defined.
1. **State the problem:** We need to find the product of the matrices
$$A = \begin{bmatrix}-9 & -6 \\ -4 & 4\end{bmatrix}$$ and $$B = \begin{bmatrix}4 \\ -5\end{bmatrix}$$ if possible.
2. **Check matrix dimensions:** Matrix $$A$$ is $$2 \times 2$$ and matrix $$B$$ is $$2 \times 1$$.
The product $$AB$$ is defined if the number of columns of $$A$$ equals the number of rows of $$B$$.
Here, $$A$$ has 2 columns and $$B$$ has 2 rows, so the product is defined.
3. **Formula for matrix multiplication:**
If $$A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$$ and $$B = \begin{bmatrix}x \\ y\end{bmatrix}$$, then
$$AB = \begin{bmatrix}ax + by \\ cx + dy\end{bmatrix}$$
4. **Calculate each element:**
Top element:
$$(-9)(4) + (-6)(-5) = -36 + 30 = -6$$
Bottom element:
$$(-4)(4) + (4)(-5) = -16 - 20 = -36$$
5. **Write the product matrix:**
$$AB = \begin{bmatrix}-6 \\ -36\end{bmatrix}$$
6. **Conclusion:** The product is defined and equals
$$\boxed{\begin{bmatrix}-6 \\ -36\end{bmatrix}}$$.