Mathematics

Problem

Let

\triangle ABC

have vertices

A(0,0)

B(1,2)

and

C(3,1)

. The coordinates of

A

B

and

C

are transformed by the matrix

\begin{pmatrix}2 & 1\\0 & 3\end{pmatrix}

to obtain

\triangle A'B'C'

. Without finding the new coordinates individually, find the ratio

\dfrac{\text{area of }\triangle A'B'C'}{\text{area of }\triangle ABC}