# Thread: Similarity and Digonalization in matrices

1. ## Similarity and Digonalization in matrices

a) If A and B are invertible matrices show that AB and BA are similar

b) Let A be an invertible matrix. Prove that if A is digonalizable, so is its inverse.(A^-1)

Thank You

2. Originally Posted by somestudent2
a) If A and B are invertible matrices show that AB and BA are similar
$BA = A^{-1}(AB)A$.

Originally Posted by somestudent2
b) Let A be an invertible matrix. Prove that if A is digonalizable, so is its inverse.(A^-1)
If A is diagonalisable then $P^{-1}AP = D$. What happens when you take the inverse of both sides of that equation?

3. Hi

for the second one by taking inverse on both sides we arive at

P^-1A^-1P = D^-1 ?

And this proves it (does it matter that Diagonal matrix is inversed now)? Or I need to make some additional calculations?

Thanks

4. Originally Posted by somestudent2
for the second one by taking inverse on both sides we arive at

P^-1A^-1P = D^-1 ?

And this proves it (does it matter that Diagonal matrix is inversed now)? Or I need to make some additional calculations?
If a diagonal matrix has an inverse then the inverse is also diagonal. So all you need to do is to explain why D is invertible. (Reason: the diagonal elements of D are the eigenvalues of A and, since A is invertible, none of them can be zero.)