`{:[(7,1,2),(9,2,1)]:}{:[(3),(4),(5)]:}+2{:[(4),(2)]:}` is equal to

Options

A. `{:[(45),(44)]:}`
B. `{:[(43),(45)]:}`
C. `{:[(44),(43)]:}`
D. `{:[(43),(44)]:}`

Answer

To solve the given expression `{:[(7,1,2),(9,2,1)]:}{:[(3),(4),(5)]:}+2{:[(4),(2)]:}`, we will follow these steps:

### Step 1: Identify the matrices
We have three matrices:
- Matrix A:
\[ A = \begin{pmatrix} 7 & 1 & 2 \\ 9 & 2 & 1 \end{pmatrix} \]
- Matrix B:
\[ B = \begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix} \]
- Matrix C:
\[ C = \begin{pmatrix} 4 \\ 2 \end{pmatrix} \]

### Step 2: Check the feasibility of multiplication
To multiply matrices A and B, we need to check if the number of columns in A equals the number of rows in B.

- Matrix A has 3 columns (2x3) and Matrix B has 3 rows (3x1).
- Since the number of columns in A (3) equals the number of rows in B (3), the multiplication is feasible.

### Step 3: Perform the multiplication \( A \times B \)
Now we will multiply matrix A by matrix B:
\[ A \times B = \begin{pmatrix} 7 & 1 & 2 \\ 9 & 2 & 1 \end{pmatrix} \times \begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix} \]

Calculating the first element:
\[ (7 \times 3) + (1 \times 4) + (2 \times 5) = 21 + 4 + 10 = 35 \]

Calculating the second element:
\[ (9 \times 3) + (2 \times 4) + (1 \times 5) = 27 + 8 + 5 = 40 \]

Thus,
\[ A \times B = \begin{pmatrix} 35 \\ 40 \end{pmatrix} \]

### Step 4: Multiply matrix C by 2
Now we will calculate \( 2C \):
\[ 2C = 2 \times \begin{pmatrix} 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 \times 4 \\ 2 \times 2 \end{pmatrix} = \begin{pmatrix} 8 \\ 4 \end{pmatrix} \]

### Step 5: Add the results from Step 3 and Step 4
Now we will add the results of \( A \times B \) and \( 2C \):
\[ \begin{pmatrix} 35 \\ 40 \end{pmatrix} + \begin{pmatrix} 8 \\ 4 \end{pmatrix} = \begin{pmatrix} 35 + 8 \\ 40 + 4 \end{pmatrix} = \begin{pmatrix} 43 \\ 44 \end{pmatrix} \]

### Final Result
Thus, the final result of the expression is:
\[ \begin{pmatrix} 43 \\ 44 \end{pmatrix} \]