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!

 

[member 587159]

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.