Using a determinant to find the area of the triangle (deriving the formula)

[Sunwoo Bae]

Homework Statement:

Shown in the text

Relevant Equations:

Det(A) = det(A^T) determinant of transpose equals determinant of the original matrix

This is the question. The following is the solutions I found:

I understand that the first line was derived by setting one vertex on origin and taking the transpose of the matrix. However, I cannot understand where the extra row and column came from in the second line. Can anyone explain how the formula is derived?

Thank you!

 

This is due to the fact that
$$\det \begin{pmatrix}a & b & 1 \\ c & d & 0 \\ e & f & 0\end{pmatrix}= \det \begin{pmatrix} c & d \\ e & f\end{pmatrix}$$

To see this, develop the $3\times 3$-determinant along the third column (cofactor expansion), or if you don't know about this calculate both sides and see that they are equal.