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. (

Algebra ->  Matrices-and-determiminant -> 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. (      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
a) If A is idempotent, then A%5E2=A. You can break this down to say that A%2AA=A. From there, left multiply both sides by A%5E%28-1%29 (you can right multiply both sides by A%5E%28-1%29 also) to get A%5E%28-1%29A%2AA=A%5E%28-1%29%2AA which then becomes I%2AA=I which simplifies to A=I


b) The matrix A is idempotent when A%5E2=A. In order for A%5E2=A to be true, A%5E2=A%2AA must be defined (ie possible). So in order for A%2AA to be defined, A must have the same number of rows and columns. This means that A must be square.


c) Does B%5E2=B ? If so, then matrix B is idempotent. Notice how . So this shows us that B%2AB=B and that B%5E2=B. So matrix B is idempotent.