Question

1, and 6. An n xn matrix A is called idempotent if A2 = A. Some examples include lude [22] fool the identity In: Idempotents correspond to "projections onto a subspace," as we will discuss later. Prove the following statements: a) If A is idempotent then so is A". b) If A is idempotent, then so is In - A. c) If A and B are both idempotent, and AB = BA= Onxn (the zero matrix), then A+B is idempotent. d) If A idempotent and nonsingular, then A=In. e) If A is idempotent and Ax = sx for some scalar s and some vector x = 0, then s= 0 or s = 1.

Answer 1

C'lempotent matrin be ut A A. then idempotent mahix, a). A S ve Then (AT)2 (a)T )E (A)T (AT)= =) AT, is also an idempotent marix. b). Let an (dempoteut mahix, ie A즌 A. Then (In- A)?= ( In- A). (Inr A) In ( In-A)- A (In A) - In - Ini A - A.Int A? - In- A -A+A AA %3D ) (In-A)- In -A:

=> (In-a) also an idempotent. B are idempotent marix A. and ut AB BA= Onxn' uro matri of Onxn s where order Then, (A PB)" = (A+B). CAPB) (A ACA+B) tB(A+ B) A Bt B. AtB.B. A.A + A2+ ABt BA +B2 AB=BA=0) using Atoto+B A + B. %3D idempotent provided (A +B) Hene is that A & Bre idempotent sE AB- BA= Onnn'

mahix an édempotent d). be Let 8 ingular. it that Such le. A A & det CA)#O. def A z R. et it possible =) det (A?) = det (A) det (A). detCA)= det(A) k'=R. => =) =) R2011 ie det (A)=0 or det (A)-1. But here bingular, det (A)=I By difinitron. A A-0 =) AlA- In)20 =) A-In=0 or " A is ) A-In =0 singular, A= In non-