Question

Please answer all parts with full & clear solutions so I can understand :)

5. If B and C are square matrices, then prove the following properties: (a) If B and C commute, then Bect = et B. (b) eCBC-? = CeBC-1.

Answer 1

Mnxn (R) or Xe Muxnck) Ans. - For xe is delined as, matrix exponential X + 4 , Xo = Inan. K=o Let B, C e Mnan (R) , Dt. BC = CB. Then then | Bect - B ( 14 (cast) : 5 * 1 ) K-O

= (Bck) — - K=o we wove by induction Bc = chB & nEN. F2 nel BC = CB a true and given. Bcmi = mtB. - 1) Let Fr n=mi, claim - Bm = comb 2) BCmt = cm1 B. Bonec=cm. (B c.) Bcm=cmi (CB) (BC=CB) Bcm=cmb.. - Hora Bcn=chbt new, By **),») Bect a cat (CAB) ko ki = (eet) (B).

(b). CBCI e CBC" e 1 (CACIJk. Ko For any ke solun, (CBCT)* = CBk ol... Easily can be preoved by induction. (CBC) = Inan. And c Boct=c Inan C"=cct = Inan Fq ko. Induction Hylothon - Let For k =m- (CBCT) m = comect - 2) claim - (CBCT) m = CBMcl. By 2) (CB ct) = C Bmlet so, a CCB (A) M = (CBCT) M (CB ct) = (C BMW ct) CC BCT) = CBM1 (64c) BCt = comat.

Hana prooved. Do. BCBt CBCT Herce pwoved., so, e C Bkct K=o - co the BK) CT =ce Bct. K=o