If U is same as in Example 50, then the value of `{:[(3,2,0)]U[(3),(2),(0)]=:}`

Options

A. 5
B. `5//2`
C. 4
D. `3//2`

Explanation

To solve the problem, we need to multiply a row matrix by a given matrix \( U \) and then multiply the result by a column matrix. Let's break down the steps:

### Step 1: Define the matrices

Let \( U \) be defined as follows:

\[ U = \begin{pmatrix} 1 & -2 & 1 \\ 2 & -1 & -4 \\ 2 & -1 & -3 \end{pmatrix} \]

The row matrix is:

\[ R = \begin{pmatrix} 3 & 2 & 0 \end{pmatrix} \]

And the column matrix is:

\[ C = \begin{pmatrix} 3 \\ 2 \\ 0 \end{pmatrix} \]

### Step 2: Multiply the row matrix \( R \) by the matrix \( U \)

To find \( R \cdot U \), we perform the multiplication as follows:

\[ R \cdot U = \begin{pmatrix} 3 & 2 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 & -2 & 1 \\ 2 & -1 & -4 \\ 2 & -1 & -3 \end{pmatrix} \]

Calculating each element of the resulting matrix:

1. First element:
\[ (3 \cdot 1) + (2 \cdot 2) + (0 \cdot 2) = 3 + 4 + 0 = 7 \]

2. Second element:
\[ (3 \cdot -2) + (2 \cdot -1) + (0 \cdot -1) = -6 - 2 + 0 = -8 \]

3. Third element:
\[ (3 \cdot 1) + (2 \cdot -4) + (0 \cdot -3) = 3 - 8 + 0 = -5 \]

Thus, we have:

\[ R \cdot U = \begin{pmatrix} 7 & -8 & -5 \end{pmatrix} \]

### Step 3: Multiply the result by the column matrix \( C \)

Now we need to multiply the resulting row matrix by the column matrix \( C \):

\[ \begin{pmatrix} 7 & -8 & -5 \end{pmatrix} \cdot \begin{pmatrix} 3 \\ 2 \\ 0 \end{pmatrix} \]

Calculating this gives:

1. First element:
\[ (7 \cdot 3) + (-8 \cdot 2) + (-5 \cdot 0) = 21 - 16 + 0 = 5 \]

### Final Result

Thus, the value of \( \{[(3,2,0)]U[(3),(2),(0)]=:\} \) is:

\[ \boxed{5} \]