Answer
11
Work Step by Step
Step 1. Identify the coordinates of the vertices of the triangle:
$(-2,-2), (-1,3),(3,1)$
Step 2. Use the formula (in section 10.6) to set up the determinant:
$\begin{array} \\A=\pm\frac{1}{2}\\ \end{array}
\begin{vmatrix} -2&-2&1\\ -1&3&1\\3&1&1\end{vmatrix}$
Step 3. Perform the row operations $R_2-R_1\to R_2$ and $R_3-R_1\to R_3$
$\begin{array} \\A=\pm\frac{1}{2}\\ \end{array}
\begin{vmatrix} -2&-2&1\\ 1&5&0\\5&3&0\end{vmatrix}$
Step 4. Evaluate the determinate using column-3 expansion:
$A=\pm\frac{1}{2}(1\times3-5\times5)=\pm11$
Step 5. Conclusion: the area of the triangle is $11$
