If `A=[(1,2,8)],B=[(4),(5),(6)]`, and `C=[(0,2,0)]` and `D=[(5),(2),(4)]`, then AB + CD is equal to

Options

A. [66]
B. [-66]
C. [32]
D. [20]

Answer

To solve the problem, we need to calculate \( AB + CD \) where:

- \( A = \begin{pmatrix} 1 & 2 & 8 \end{pmatrix} \) (1x3 matrix)
- \( B = \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix} \) (3x1 matrix)
- \( C = \begin{pmatrix} 0 & 2 & 0 \end{pmatrix} \) (1x3 matrix)
- \( D = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} \) (3x1 matrix)

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

To multiply matrices \( A \) and \( B \):

\[ AB = \begin{pmatrix} 1 & 2 & 8 \end{pmatrix} \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix} \]

Using the formula for matrix multiplication, we compute:

\[ AB = 1 \cdot 4 + 2 \cdot 5 + 8 \cdot 6 \]

Calculating each term:

\[ = 4 + 10 + 48 \]

Adding these values together:

\[ = 62 \]

Thus,

\[ AB = \begin{pmatrix} 62 \end{pmatrix} \quad \text{(1x1 matrix)} \]

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

Next, we calculate \( CD \):

\[ CD = \begin{pmatrix} 0 & 2 & 0 \end{pmatrix} \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} \]

Using the formula for matrix multiplication, we compute:

\[ CD = 0 \cdot 5 + 2 \cdot 2 + 0 \cdot 4 \]

Calculating each term:

\[ = 0 + 4 + 0 \]

Adding these values together:

\[ = 4 \]

Thus,

\[ CD = \begin{pmatrix} 4 \end{pmatrix} \quad \text{(1x1 matrix)} \]

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

Now we add the results of \( AB \) and \( CD \):

\[ AB + CD = \begin{pmatrix} 62 \end{pmatrix} + \begin{pmatrix} 4 \end{pmatrix} \]

Calculating the sum:

\[ = 62 + 4 = 66 \]

Thus,

\[ AB + CD = \begin{pmatrix} 66 \end{pmatrix} \quad \text{(1x1 matrix)} \]

### Final Answer

The final result is:

\[ AB + CD = 66 \]

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