Area of a linearly transformed curve

Geometry Level 4

Let \( C \) be a closed curve in the \( xy \) plane, with an enclosed area of \( A \). The points on this curve are operated on by a linear transform, given by \( \mathbf{y = T x} \), where \( \mathbf{x} \) is the vector representing any point on \( C \), and \( \mathbf{y} \) is the vector of the its image, and matrix \( T \) is given by

\( T = \begin{bmatrix} 1 && 2 \\ 3 && 4 \end{bmatrix} \)

What will be the area of the transformed curve?

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