The matrix producet
`[(1),(-2),(3)][(4,5,2)][(2),(-3),(5)]` equals

Options

A. `[(3),(-6),(9)]`
B. `[(3),(6),(9)]`
C. `[(3),(6),(-9)]`
D. None of these

Answer

To find the product of the matrices given, we will follow the matrix multiplication rules step by step.

### Step 1: Define the Matrices
Let:
- Matrix A = \(\begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}\)
- Matrix B = \(\begin{pmatrix} 4 & 5 & 2 \end{pmatrix}\)
- Matrix C = \(\begin{pmatrix} 2 \\ -3 \\ 5 \end{pmatrix}\)

### Step 2: Multiply Matrices B and C
We will first multiply Matrix B and Matrix C. The multiplication of a \(1 \times 3\) matrix with a \(3 \times 1\) matrix results in a \(1 \times 1\) matrix.

\[ B \cdot C = \begin{pmatrix} 4 & 5 & 2 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ -3 \\ 5 \end{pmatrix} \]

Calculating this step-by-step:
- The first element: \(4 \cdot 2 = 8\)
- The second element: \(5 \cdot (-3) = -15\)
- The third element: \(2 \cdot 5 = 10\)

Now, we sum these results:
\[ 8 + (-15) + 10 = 8 - 15 + 10 = 3 \]

So, \(B \cdot C = \begin{pmatrix} 3 \end{pmatrix}\).

### Step 3: Multiply Matrix A with the Result from Step 2
Now we will multiply Matrix A with the result from the previous multiplication, which is a \(1 \times 1\) matrix.

\[ A \cdot (B \cdot C) = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix} \cdot \begin{pmatrix} 3 \end{pmatrix} \]

This multiplication will yield a \(3 \times 1\) matrix:
\[ = \begin{pmatrix} 1 \cdot 3 \\ -2 \cdot 3 \\ 3 \cdot 3 \end{pmatrix} = \begin{pmatrix} 3 \\ -6 \\ 9 \end{pmatrix} \]

### Final Result
The final product of the matrices is:
\[ \begin{pmatrix} 3 \\ -6 \\ 9 \end{pmatrix} \]