Question

10 the area of a triangle with vertices
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 $\

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

$

Explanation:

Step1: Identify \(a\), \(b\), \(c\), \(d\)

Given matrix \(

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

\), so \(a = - 4\), \(b = 7\), \(c = 6\), \(d=-10\).

Step2: Calculate \(ad - bc\)

$$
LATEXBLOCK1
$$

Step3: Find the inverse matrix

The inverse matrix formula is \(\frac{1}{ad - bc}

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

\).
Substitute \(a=-4\), \(b = 7\), \(c = 6\), \(d=-10\) and \(ad - bc=-2\) into the formula:

$$
LATEXBLOCK3
$$

Answer:

D. \(

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

\)