Why is it that there is a one (1) entry in the third column of a 3 x 3 matrix in solving or?

Question

finding the area of a triangular area with given coordinates of three points of (x,y)? Could it not be solved by using a 2 x 2 matrix?

like x1 y1
x2 y2
x3 y3

like

x1 y1
x2 y2
x3 y3

Answer 1

It's clear that you can't stuff it into a 2x2 matrix since you have 6 terms. However, you could have a series of 2x2 matrices if you liked. But this whole way of getting area is more general, and the fact that when you have three points you can show it as a determinant is an artifact.

If you have (in counter clockwise order) points (x1,y1), (x2, y2), ... (xn, yn), then the area will be one half of:
x1*y2 + x2*y3 + x3*y4 + x4*y5 + ... + xn*y1 - (
y1*x2 + y2*x3 + y3*x4 + y4*x5 + ... + yn*x1)

You can write this down as two colums of numbers with x's in the left and y's in the right. Now imagine lines connecting the pairs of numbers that you have to multiply and spot the pattern (the downward sloping lines represent + multiplications, the upward sloping lines represent - terms). This is a really useful technique to remember. It also shows that if you have integer coefficients, your area will be a multiple of 1/2. So it is just fortuitous that with three points you wind up with a 3 by 3 determinant.

Answer 2

You said "triangular area with given coordinates of three points of (x,y)?" Did you mean to have a variable z as well seeing as a triangle has three sides and 3 points? That would be why it is a 3x3 matrix, you need to put in z as well, not just x and y.

Answer 3

The reason for having a three by three matrix is because you will have a variable z later on and that is for objects that exist in a three dimensional plain.

Answer 4

no it can not.
To define a triangle you will need 3 points. There is a 1 probably to normalize the points , by dividing all popints with the max value .