Question

10 the area of a triangle with vertices at the points $(a,b),(c,d)$, and $(e,f)$ is $\frac{1}{2}|\det a|$, where $a = \

$$\begin{bmatrix} a&b&1 \\\\ c&d&1 \\\\ e&f&1 \\end{bmatrix}$$

$. if $\det a=ad + be+cf - ed - af - bc$ and the vertices of a triangle are at the points $(-7,5),(3,-6)$, and $(1,9)$, what is the area of the triangle? f $-64$ g $-32$ h $32$ j $64$ k $128$ 11 the inverse of the matrix $\

$$\begin{bmatrix} a&b \\\\ c&d \\end{bmatrix}$$

$ is $\frac{1}{ad - bc}\

$$\begin{bmatrix} d&-b \\\\ -c&a \\end{bmatrix}$$

$. what is the inverse of $\

$$\begin{bmatrix} -4&7 \\\\ 6&-10 \\end{bmatrix}$$

$? a $\

$$\begin{bmatrix} -10&-7 \\\\ -6&-4 \\end{bmatrix}$$

$ b $\

$$\begin{bmatrix} 10&7 \\\\ 6&4 \\end{bmatrix}$$

$ c $\

$$\begin{bmatrix} -5&-\\frac{7}{2} \\\\ -3&-2 \\end{bmatrix}$$

$. d $\

$$\begin{bmatrix} 5&\\frac{7}{2} \\\\ 3&2 \\end{bmatrix}$$

$ e $\

$$\begin{bmatrix} 5&7 \\\\ 3&2 \\end{bmatrix}$$

$ 12 $\

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

\times\

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

=$ f $\

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

$ g $\

$$\begin{bmatrix} 1&1 \\\\ 0&0 \\end{bmatrix}$$

$ h $\

$$\begin{bmatrix} 10&-1 \\\\ 0&0 \\end{bmatrix}$$

$ j $\

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

$ k $\

$$\begin{bmatrix} 10&-1 \\\\ -2&0 \\end{bmatrix}$$

$

Explanation:

Step1: Identify the values of \(a,b,c,d,e,f\)

Given the vertices \((-7,5)\), \((3,-6)\), and \((1,9)\), we have \(a=-7\), \(b = 5\), \(c = 3\), \(d=-6\), \(e = 1\), \(f=9\).

Step2: Calculate \(detA\)

Substitute the values into \(detA=ad+be+cf - ed - af - bc\):

$$
LATEXBLOCK0
$$

Step3: Calculate the area of the triangle

The area of the triangle is \(\frac{1}{2}|detA|\).

$$
LATEXBLOCK1
$$

Answer:

J. 64