Question

1. add the matrices.

\\(\

$$\begin{bmatrix} 5 & 2 & 1 & -4 \\\\ 1 & -1 & 3 & 0 \\\\ -3 & 0 & -2 & 3 \\end{bmatrix}$$

+ \

$$\begin{bmatrix} -2 & -2 & -5 & 2 \\\\ -4 & 0 & -7 & 0 \\\\ 3 & -6 & 4 & -3 \\end{bmatrix}$$

\\)
options:
\\(\

$$\begin{bmatrix} 5 & 2 & -5 & -4 \\\\ -4 & -1 & -7 & 0 \\\\ 3 & 0 & 4 & 3 \\end{bmatrix}$$

\\)
\\(\

$$\begin{bmatrix} 3 & -3 & 0 \\\\ 0 & -1 & 0 \\\\ -4 & -4 & 2 \\\\ -2 & 0 & 0 \\end{bmatrix}$$

\\)
\\(\

$$\begin{bmatrix} 3 & 0 & -4 & -2 \\\\ -3 & -1 & -4 & 0 \\\\ 0 & 0 & 2 & 0 \\end{bmatrix}$$

\\)
\\(\

$$\begin{bmatrix} -11 \\end{bmatrix}$$

\\)

1. find the equations of the asymptotes of the hyperbola. \\(\frac{(x - 2)^2}{4^2} - \frac{(y - 1)^2}{2^2} = 1\\)

use the formula \\(y = \pm \frac{b}{a}(x - x_1) + y_1\\).
options:
\\(y = 0.5x\\), \\(y = -0.5x + 2\\)
\\(y = 0.5x - 2\\), \\(y = -0.5 + 2\\)
\\(y = 2x + 5\\), \\(y = -2x - 3\\)
\\(y = 2x - 3\\), \\(y = -2x + 5\\)

Explanation:

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Problem 16:

Step1: Add corresponding matrix entries

To add two matrices, sum the elements in the same position:

$$\begin{bmatrix} 5+(-2) & 2+(-2) & 1+(-5) & -4+2 \\ 1+(-4) & -1+0 & 3+(-7) & 0+0 \\ -3+3 & 0+(-6) & -2+4 & 3+(-3) \end{bmatrix}$$

Step2: Simplify each entry

Calculate the arithmetic for each position:

$$\begin{bmatrix} 3 & 0 & -4 & -2 \\ -3 & -1 & -4 & 0 \\ 0 & -6 & 2 & 0 \end{bmatrix}$$

Step1: Identify hyperbola parameters

From $\frac{(x-2)^2}{4^2} - \frac{(y-1)^2}{2^2}=1$, we get $(x_1,y_1)=(2,1)$, $a=4$, $b=2$.

Step2: Substitute into asymptote formula

Use $y=\pm\frac{b}{a}(x-x_1)+y_1$:
First asymptote: $y=\frac{2}{4}(x-2)+1 = 0.5x -1 +1 = 0.5x$
Second asymptote: $y=-\frac{2}{4}(x-2)+1 = -0.5x +1 +1 = -0.5x +2$

Answer:

$$\begin{bmatrix} 3 & 0 & -4 & -2 \\ -3 & -1 & -4 & 0 \\ 0 & -6 & 2 & 0 \end{bmatrix}$$

(matches the third option)

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Problem 17: