Linear Algebra- Diagonal Matrix

[Roni1985]

Homework Statement

Let D be an n x n diagonal matrix whose diagonal entries are either 0 or 1

a) Show that D is idempotent

b) Show that if X is a nonsingular matrix and A=XD(X)^-1 , then A is idempotent

Homework Equations

The Attempt at a Solution

a) I tried it, and it works for a specific matrix, say 3x3... but I don't know if it's really a proof.
I need to show that

D²=D

b) I solved this part

Also,
is (AB)²= A²*B²
or
(AB)²=B²*A²
?

 

[Dick]

You just need to make up some notation. Let diag(a1,a2,...an) be the matrix whose diagonal entries are a1, a2... Now what diag(a1,a2,...an)*diag(b1,b2,...bn)? (AB)^2 generally isn't equal to either A^2*B^2 or B^2*A^2. It's just ABAB.

 

[Roni1985]

You just need to make up some notation. Let diag(a1,a2,...an) be the matrix whose diagonal entries are a1, a2... Now what diag(a1,a2,...an)*diag(b1,b2,...bn)? (AB)^2 generally isn't equal to either A^2*B^2 or B^2*A^2. It's just ABAB.

oh, right , it works ...
thanks a lot for the help ...