Solving for A ( with inverses and transpose)

Question

(2A^T + 3T)^-1 = [-4 3 ]

                      [ 1 -1 ] 

 

Please show me steps. I dont understand how to undo the inverse and the transpose and get rid of the 2 and 3

Answer 1

I will show you the steps, and you can fill in the math.  Let B = the matrix on the right hand side.  Note that 

 

det(B) ≠ 0,

 

so B^-1 exists.

 

(2A^T + 3T)^-1 = B

 

Take inverse on both sides and recall that (M^-1)^-1 = M

 

[(2A^T + 3T)^-1]^-1 = B^-1

 

The left hand side is just the inverse of the inverse, so

 

2A^T + 3T = B^-1

 

2A^T = B^-1 - 3T

 

A^T = (1/2)(B^-1 - 3T)

 

Now take the transpose on both sides and remember that (A^T)^T = A

 

A = (1/2)(B^-1 - 3T)^T

 

This at least gets you as far as A in terms of T and B.