Finding Linear transformation, that transforms original point in addition to consecutive images . Original Question
I have inferred from the question that a linear transformation T, originally transforming point (1,3), also transforms its image(-2,-3) in adittion to the image following that(2,4). However when attempting to find the matrix trasnforming (1,3) to (-2,-3), i found it was inconsistant with that which transformed (-2,-3) to (2,4). So i am led to conclude thati have not understood the question correctly.
Attempted Working: T(X) = X' .  T=$\begin{bmatrix}a & b\\c & d\end{bmatrix}$
$\begin{bmatrix}a & b\\c & d\end{bmatrix}$·$\begin{bmatrix}1\\3\end{bmatrix}$ = $\begin{bmatrix}a +3b\\c+3d\end{bmatrix}$ $=>$ $a+3b = -2$ & $c+3d=-3$ 
$\begin{bmatrix}a & b\\c & d\end{bmatrix}$·$\begin{bmatrix}-2\\-3\end{bmatrix}$ = $\begin{bmatrix}-2a -3b\\-2c-3d\end{bmatrix}$ $=>$ $-2a-3b = 2$ & $-2c-3d=4$ 
$\begin{bmatrix}a & b\\c & d\end{bmatrix}$·$\begin{bmatrix}2\\4\end{bmatrix}$ = $\begin{bmatrix}2a +4b\\2c+4d\end{bmatrix}$ $=>$ $2a+4b = -3$ & $2c+4d=-11$ 
There shouldn't be any x or y values that satisfy all 3 or else an equation is redundant according to https://www.physicsforums.com/threads/three-equations-two-unknowns.426629/
, so i'm just wondering how i'm interpreting it wrong. The answer is  1/2$\begin{bmatrix}-1 & -1\\-21 & 5\end{bmatrix}$ according to my textbook. 

It seems there is a typo in the text, it should be
$$\begin{bmatrix}a & b\\c & d\end{bmatrix} \begin{bmatrix}1\\3\end{bmatrix}=\begin{bmatrix}-2\\-3\end{bmatrix}$$
$$\begin{bmatrix}a & b\\c & d\end{bmatrix} \begin{bmatrix}2\\4\end{bmatrix}=\begin{bmatrix}-3\\-11\end{bmatrix}$$
which can be solved to obtain the answer.
