Word problem help involving matrices?

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wrongnmbr

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A real estate agent is writing a listing for a triangular piece of land. She has to include the number of square feet for the property and has to calculate it from a plot that shows the following information: one corner of the plot is 152 feet south and 136 feet east from the upper vertex of the plot, the other corner is 36 feet south and 240 feet east from the upper vertex of the plot. Use matrices to find the area of the piece of land.

Okay, now I'm aware that my points are (0,0), (-152, 136) and (240, -36), but I'm confused as to how I would find the area using matrices. Help?
 
You use the deteminant. Given the points (A,B), (C,D), and (E,F):

The area of a triangle can be found by using \(\displaystyle \frac{1}{2}\begin{vmatrix}A&B&1\\C&D&1\\E&F&1\end{vmatrix}\)

That's 1/2 times the determinant of the matrix made up of the coordinates given.

In other words, \(\displaystyle \frac{1}{2}\begin{vmatrix}0&0&1\\-36&240&1\\-152&136&1\end{vmatrix}\)

Check against Heron's formula.
 
Okay, in lieu of Heron's formula, we've just been inserting the matrices into our TI-83s, so in this scenario I just inserted:

\(\displaystyle \frac{1}{2}\begin{vmatrix}0&0&1\\-36&240&1\\-152&136&1\end{vmatrix}\)

...into my calculator, and I came up with...

\(\displaystyle \begin{vmatrix}0&0&.5\\-18&120&.5\\-76&68&.5\end{vmatrix}\)

How would I find the area from this point? Sorry, I'm practically lost here...
 


You need to calculate the determinant of the matrix. That's a number, not a matrix. In other words, you multiply 1/2 times the determinant.

Did they teach you in class how to ask the calculator for the matrix determinant?

Also, note that galactus accidently transposed the values for C and D, in his matrix. Are you paying attention to the meaning of the symbols, in his post? I'm wondering because you're using the wrong matrix.

 
Yeah, looking back I think he meant to write it as...

\(\displaystyle \begin{vmatrix}0&0&1\\-152&136&1\\240&-36&1\end{vmatrix}\)

And if I'm understanding you correctly, I now need to find the determinant of this matrix and multiply it by 1/2? If I'm still off, we're probably on different pages...
 
\(\displaystyle A \ = \ \frac{1}{2}[(0)(240)+(-36)(136)+(-152)(0)-(0)(130)-(-152)(240)-(-36)(0)] \ = \ 15,792 \ sq. \ ft.\)
 
wrongnmbr said:
Yeah, looking back I think he meant to write it as...

\(\displaystyle \begin{vmatrix}0&0&1\\-152&136&1\\240&-36&1\end{vmatrix}\)

And if I'm understanding you correctly, I now need to find the determinant of this matrix and multiply it by 1/2? If I'm still off, we're probably on different pages...
Click to expand...


Looks to me like it should be:

\(\displaystyle A=\frac{1}{2}\begin{vmatrix}0&0&1\\136&-152&1\\240&-36&1\end{vmatrix}\)
 
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