Effect of a linear transformation on the perimeter of a shape

let assume $A: R^2 \rightarrow R^2$ is a linear transformation. (A is $2 \times 2$ matrix)

We know that the area of a shape's image is $ |det(A)| * S$, which $S$ is our original shape's area.

And my question is that what about the perimeter? Is there any relation between the original shape's perimeter and its image's perimeter?

I tried some examples, but I could not find a general relation!

Thank you

The perimeter of the image of a shape can be arbitrary large even with a matrix with determinant $1$.

Take for example the image of unit square $(0,0),(1,0),(1,1),(0,1)$ by transvection with matrix

$$\begin{pmatrix}1&m\\0&1\end{pmatrix}$$

which is the parallelogram $(0,0),(0,1),(m+1,1),(m,1)$ with arbitrary large perimeter:

$$2+2 \sqrt{1+m^2}$$

Fig. 1: The case $m=5$.