If `A = [ (2,-3,4)], B = [(3),(2),(2)], x = [(1,2,3)] and Y = [(2),(3),(4)]` then AB + XY equals

Options

A. [28]
B. [24]
C. 28
D. 24

Answer

To solve the problem, we need to calculate \( AB + XY \) step by step.

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

Given:
- \( A = \begin{pmatrix} 2 & -3 & 4 \end{pmatrix} \) (1 x 3 matrix)
- \( B = \begin{pmatrix} 3 \\ 2 \\ 2 \end{pmatrix} \) (3 x 1 matrix)

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

Using the formula for matrix multiplication:
\[ AB = (2 \cdot 3) + (-3 \cdot 2) + (4 \cdot 2) \]
Calculating each term:
\[ = 6 - 6 + 8 = 8 \]

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

Given:
- \( X = \begin{pmatrix} 1 & 2 & 3 \end{pmatrix} \) (1 x 3 matrix)
- \( Y = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} \) (3 x 1 matrix)

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

Using the formula for matrix multiplication:
\[ XY = (1 \cdot 2) + (2 \cdot 3) + (3 \cdot 4) \]
Calculating each term:
\[ = 2 + 6 + 12 = 20 \]

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

Now, we add the results from Step 1 and Step 2:
\[ AB + XY = 8 + 20 = 28 \]

### Final Answer:
\[ AB + XY = 28 \]

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