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

  • #1
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Homework Statement:

Shown in the text

Relevant Equations:

Det(A) = det(A^T) determinant of transpose equals determinant of the original matrix
7AD7CFA7-C0E0-47CF-9FDA-EE6343366B2C.jpeg

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

D91C0495-3838-4DFA-9EAE-B1A8928292BA.jpeg

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!
 

Answers and Replies

  • #2
Math_QED
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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.
 
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