Inverse of the matrix of a linear transformation

[alias]

Homework Statement

Let T: M22→M22 be a LT defined by T(A)=AB where
B=[3,2
2,1]
Determine if T is invertible with respect to standard bases B=C={e11,e12,e21,e22}. If so, use (equation below) to find T^-1.

Homework Equations

[[T^-1 [AB]]_C = [T^-1]_{B to C matrix} [AB]_B (at least I think this is the right one)

The Attempt at a Solution

I found:
T= [ [3,2,0,0]’,[2,1,0,0]’,[0,0,3,2]’,[0,0,2,1]’ ]
T^-1 = [ [-1,2,0,0]’,[2,-3,0,0]’,[0,0,-1,2]’,[0,0,2,-3]’ ]
My TA told me I was doing this wrong and I can't figure out what to do. Can anyone help?

 

[lanedance]

so if get it right, if T is invertible then
$$T^{-1}(AB) = (AB)B^{-1} = A$$

so isn't it sufficent to find whether B has an inverse

 

[alias]

Much appreciated