Question

If $ X = A \times B $, $ A = \begin{bmatrix} 1 & 2 \\-1 & 1 \end{bmatrix} $, $ B = \begin{bmatrix} 3 & 6 \\5 & 7 \end{bmatrix} $, find $ x_1 + x_2 $.

• A. \( 12 \)
• B. \( 33 \)
• C. \( 10 \)
• D. \( 8 \)

Hint:

When multiplying matrices, always follow the row-by-column multiplication rule for each element. The resulting matrix will contain the sums of products of corresponding entries.

Answer 1

The Correct Option is B

Let $X = A \times B$, where $A = \begin{bmatrix} 1 & 2 \\-1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 6 \\5 & 7 \end{bmatrix}$.

To find the matrix $X$, we multiply the matrices $A$ and $B:
$$ X = \begin{bmatrix} 1 & 2 \\-1 & 1 \end{bmatrix} \times \begin{bmatrix} 3 & 6 \\5 & 7 \end{bmatrix} = \begin{bmatrix} (1)(3) + (2)(5) & (1)(6) + (2)(7) \\(-1)(3) + (1)(5) & (-1)(6) + (1)(7) \end{bmatrix} $$
$$ X = \begin{bmatrix} 3 + 10 & 6 + 14 \\-3 + 5 & -6 + 7 \end{bmatrix} = \begin{bmatrix} 13 & 20 \\2 & 1 \end{bmatrix} $$

Let $X = \begin{bmatrix} x_1 & x_2 \\x_3 & x_4 \end{bmatrix} = \begin{bmatrix} 13 & 20 \\2 & 1 \end{bmatrix}$.
Then $x_1 = 13$ and $x_2 = 20$.
We want to find $x_1 + x_2$, so $x_1 + x_2 = 13 + 20 = 33$.

Final Answer: The final answer is $\boxed{33}$