Inverse of a linear transformation

[stunner5000pt]

find the inverse of $$T \left[ \begin{array}{cc} a&b \\ c&d \end{array} \right] = \left[ \begin{array}{cc} a+2c&b+2d \\ 3c-a&3d-b \end{array} \right]$$

do i row reduce the transformation matrix... it doesn work , though
is there an easier way??

 

[0rthodontist]

You have to recognize that writing a b c d as a matrix like that is similar to writing a b c d as a column vector. The _real_ matrix of the transformation is 4x4.

 

[HallsofIvy]

As Orthodontists pointed out (remarkable how good at math a dentist is!) since this maps a 2 by 2 matrix into a 2 by 2 matrix, the actual "transformation matrix" is 4 by 4. If you don't want to write all of that out, try a more basic approach. Using p,q,r,s as the components of the result of the transformation, you have four equations:
p= a+ 2c, q= b+ 2d, r= 3c- a, s= 3d- b, four equation in the four "unknowns" a, b, c, d. Solve for a, b, c, d in terms of p,q,r,s. that will give the equations for the inverse transformation.