# Inverse of the matrix of a linear transformation

1. Jul 14, 2010

### alias

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

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

3. 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?

Last edited: Jul 14, 2010
2. Jul 15, 2010

### 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

3. Jul 16, 2010

### alias

Much appreciated