SOLUTION: (a)If A is idempotent and has an inverse A^(-1) show that A=I Explain what this result says about the kinds of matrices that could possibly be both invertible and idempotent. (

Question 350511: (a)If A is idempotent and has an inverse A^(-1) show that A=I
Explain what this result says about the kinds of matrices that could possibly be both invertible and idempotent.
(b) If A is idempotent, does it have to be square? Why or why not?
(c) Let B=[1 0
0 0] Is B idempotent?

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
a) If A is idempotent, then . You can break this down to say that . From there, left multiply both sides by (you can right multiply both sides by also) to get which then becomes which simplifies to

b) The matrix A is idempotent when . In order for to be true, must be defined (ie possible). So in order for to be defined, A must have the same number of rows and columns. This means that A must be square.

c) Does ? If so, then matrix B is idempotent. Notice how . So this shows us that and that . So matrix B is idempotent.
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