Geometrical meaning of the formula for the area of a triangle using a $3\times3$ matrix.

Subtract the second and the third columns by the first (this is geometrically a shear). Now you get a block lower triangular matrix and the determinant becomes the area of a parallelogram (or strictly speaking, a slanted cylinder with a parallelogram base and unit height) with two adjacent sides $\pmatrix{x_B-x_A\\ y_B-y_A}$ and $\pmatrix{x_C-x_A\\ y_C-y_A}$. Multiply it by one half, you get the area of the triangle.

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Area

Triangles

Analytic Geometry

Determinant

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