How to find the area of the image under T?

Question

For this question, I'm not sure how to continue further. Here is what I have so far. Can anyone please help me out?

Let $T:\Bbb R^2 \to \Bbb R^2$ be the linear transformation defined by $T([x,y]) = [4x-2y, 2x+3y]$. Find the area of the image under T of each of the given regions in $\Bbb R^2$.

The parallelogram determined by $2e_1 + 3e_2$ and $4e_1-e_2$.

$A = [4,-2],[2,3]$

$A^T = [4,2],[-2,3]$

$ A^T*A = [20,-2],[-2,13]$

$det([20,-2],[-2,13]) = 256$

$V = \sqrt{256} = 16$

Answer 1

Hint: The area of a region under a linear map $T$ is scaled by $|\det(T)|$. Your problem boils down to finding the area of the original parallelogram and then multiplying it by the absolute value of the determinant you found.