`[{:(3,2,1):}] [{: ( 2),(-2),(-1):}]`=______

Options

A. [2]
B. 1
C. [1]
D. 2

Answer

To solve the matrix multiplication problem given by the question `[{:(3,2,1):}] [{: ( 2),(-2),(-1):}]`, we will follow these steps:

### Step 1: Identify the matrices
We have two matrices:
- Matrix A (3x1):
\[ A = \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix} \]
- Matrix B (3x1):
\[ B = \begin{pmatrix} 2 \\ -2 \\ -1 \end{pmatrix} \]

### Step 2: Check the dimensions for multiplication
Matrix A has dimensions 3x1, and Matrix B has dimensions 3x1. However, we need to multiply a 3x1 matrix with a 1x3 matrix to perform the multiplication correctly. Thus, we will consider Matrix B as a 1x3 matrix:
\[ B = \begin{pmatrix} 2 & -2 & -1 \end{pmatrix} \]

### Step 3: Perform the multiplication
Now, we can multiply Matrix A (3x1) with Matrix B (1x3). The result will be a 3x3 matrix.

The multiplication is done as follows:
\[ C = A \times B = \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix} \times \begin{pmatrix} 2 & -2 & -1 \end{pmatrix} \]

Calculating each element of the resulting matrix C:
- \( C_{11} = 3 \times 2 = 6 \)
- \( C_{12} = 3 \times -2 = -6 \)
- \( C_{13} = 3 \times -1 = -3 \)

- \( C_{21} = 2 \times 2 = 4 \)
- \( C_{22} = 2 \times -2 = -4 \)
- \( C_{23} = 2 \times -1 = -2 \)

- \( C_{31} = 1 \times 2 = 2 \)
- \( C_{32} = 1 \times -2 = -2 \)
- \( C_{33} = 1 \times -1 = -1 \)

Thus, the resulting matrix C is:
\[ C = \begin{pmatrix} 6 & -6 & -3 \\ 4 & -4 & -2 \\ 2 & -2 & -1 \end{pmatrix} \]

### Final Answer
The result of the multiplication is:
\[ \begin{pmatrix} 6 & -6 & -3 \\ 4 & -4 & -2 \\ 2 & -2 & -1 \end{pmatrix} \]