Triangle Area with Matrices

Hello,I'm Leonardo. I want to ask something that make me curious until now. At school, we have already learned that triangle area can derived from many things, like a standard formula (half times base times height), sine formula, cosine formula, and Heron's Formula. When I try to find another way to calculate the area, I just found that we can use matrices to find the area. But, I still don't understand how can matrices help us to find the area of triangle? For example, if we want to find the area of triangle with vertices A(0,9), B(3,7), and C(2,4), what is the shortest way to find this area without draw it in the Cartesius coordinate? Is matrices formula for triangle area have any criteria? Can we use it to find another polygon area? Thanks

Easy Math Editor

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Comments

Here is a proof of the formula for the area of a triangle with determinants. It's not exactly matrices that help with finding the area, it's the related topic of determinants. For the triangle with those points above you would have to solve the following determinant: $\left| \begin{array}{ccc} 0 & 9 & 1 \\ 3 & 7 & 1 \\ 2 & 4 & 1 \end{array} \right|=11$ Then you must take the absolute value of half of it to obtain your area. The determinant is set up with the x's in the first column, the y's in the second, and 1's in the third. There is no prior criteria for this method, besides being the most useful when you have the Cartesian coordinates of the vertices. It seems like this method only works for triangles in two dimensions, for increasing the number of points makes the determinant not-square, and there is no such thing as a not-square determinant. However, it seems the method can be extended into objects with n+1 edges in n dimensions.