A=`[[2,-3,4]], B = [[3],[2],[2]], X = [[1,2,3]]` and `Y = [[2],[3],[4]]`, then AB + XY =

Options

A. [28]
B. `[[35],[36],[37]]`
C. `[[35,36,37]]`
D. `[[28,0,0],[0,28,0],[0,0,28]]`

Answer

To solve the problem \( AB + XY \), we will follow these steps:

### Step 1: Calculate \( AB \)

Given matrices:
- \( A = \begin{bmatrix} 2 & -3 & 4 \end{bmatrix} \) (1x3 matrix)
- \( B = \begin{bmatrix} 3 \\ 2 \\ 2 \end{bmatrix} \) (3x1 matrix)

To multiply \( A \) and \( B \):
\[ AB = \begin{bmatrix} 2 & -3 & 4 \end{bmatrix} \begin{bmatrix} 3 \\ 2 \\ 2 \end{bmatrix} \]

Calculating the multiplication:
\[ AB = (2 \cdot 3) + (-3 \cdot 2) + (4 \cdot 2) \]
\[ = 6 - 6 + 8 \]
\[ = 8 \]

Thus, \( AB = \begin{bmatrix} 8 \end{bmatrix} \) (1x1 matrix).

### Step 2: Calculate \( XY \)

Given matrices:
- \( X = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \) (1x3 matrix)
- \( Y = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} \) (3x1 matrix)

To multiply \( X \) and \( Y \):
\[ XY = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} \]

Calculating the multiplication:
\[ XY = (1 \cdot 2) + (2 \cdot 3) + (3 \cdot 4) \]
\[ = 2 + 6 + 12 \]
\[ = 20 \]

Thus, \( XY = \begin{bmatrix} 20 \end{bmatrix} \) (1x1 matrix).

### Step 3: Calculate \( AB + XY \)

Now we add the results from steps 1 and 2:
\[ AB + XY = \begin{bmatrix} 8 \end{bmatrix} + \begin{bmatrix} 20 \end{bmatrix} \]
\[ = 8 + 20 \]
\[ = 28 \]

Thus, the final result is:
\[ AB + XY = \begin{bmatrix} 28 \end{bmatrix} \]

### Final Answer
The answer is \( 28 \).

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