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

In summary, the formula for the determinant of a 3x3 matrix is derived by setting one vertex at the origin and taking the transpose of the matrix. The extra row and column in the second line come from the cofactor expansion of the determinant. This can also be seen by calculating both sides of the equation and seeing that they are equal.
  • #1
Sunwoo Bae
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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!
 
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  • #2
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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Related to Using a determinant to find the area of the triangle (deriving the formula)

1. How is the determinant used to find the area of a triangle?

The determinant is used to find the area of a triangle by taking the coordinates of the three vertices of the triangle and arranging them in a 3x3 matrix. The absolute value of the determinant of this matrix is then divided by 2 to get the area of the triangle.

2. What is the formula for finding the determinant of a 3x3 matrix?

The formula for finding the determinant of a 3x3 matrix is (a*d - b*c), where a, b, c, and d are the elements of the matrix arranged in a specific pattern. This formula can also be written as ad - bc.

3. How does the determinant formula relate to the area of a triangle?

The determinant formula for a 3x3 matrix is related to the area of a triangle because it represents the signed area of a parallelogram formed by the three vertices of the triangle. Dividing this value by 2 gives the area of the triangle.

4. Can the determinant be used to find the area of any triangle?

Yes, the determinant can be used to find the area of any triangle as long as the coordinates of the three vertices are known. However, the formula may need to be adjusted depending on the orientation of the triangle on the coordinate plane.

5. Are there any other methods for finding the area of a triangle?

Yes, there are other methods for finding the area of a triangle, such as using the Pythagorean theorem or the Heron's formula. However, using the determinant method is often preferred as it is more general and can be applied to any type of triangle.

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