Hey Tala.

Recall that for a linear transformation you have Ax = b where x is transformed by A to b.

In this case x and b are 2x2 matrices not vectors but since you have x and b, you can calculate A by doing A*x*x^(-1) = b*x^(-1) which means A = b*x^(-1) (x is invertible since its a 2x2 matrix).

So your solution is A = b*x^(-1) where x is your original 2x2 system and b is your final 2x2 system. Using Octave I got the solution:

>> X = [-2, -3; 5, 8]

X =

-2 -3

5 8

>> B = [2 -4; -1, 2]

B =

2 -4

-1 2

>> A = B*inv(X)

A =

-36.0000 -14.0000

18.0000 7.0000

>> A*X

ans =

2.0000 -4.0000

-1.0000 2.0000