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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-?

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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 == (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 - B(b). CBCI e CBC e 1 (CACIJk. Ko For any ke solun, (CBCT)* = CBk ol... Easily can be preoved by induction. (CBC) = Inan. AndHana prooved. Do. BCBt CBCT Herce pwoved., so, e C Bkct K=o - co the BK) CT =ce Bct. K=o

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