Question

linear algebra problem

1. True or False. If true, explain why. If false provide a counterexample. . If A? - B2, then A - B (you can assume that A and B have the same size). • If columns 1 and 3 of B are the same, so are columns 1 and 3 of AB. • If rows 1 and 3 of B are the same, so are rows 1 and 3 of AB. • (AB) - A’B?

Answer 1

a) If A = 8², The A = 8 Annwer: FALSE counter Example Let u t A=[68] B= [! :] fer, AR= [68] [8] = [60] P = [oto:] = [08] Hence, A=p3 ; but A+B AP = lep -> AP GP To (4-B ) (A+B) = 0 Either A-B=0 OR A+B=0 so, we word, A-B to, so Take A+B=0 Idea & A=-8 If columns of "1" md '3" were of "B" are some so, columns " and "3" of AB are some Answer : True.

Lel A = [aid i=1,2..n j=1,2,..n i=1,2--in j = 1,2--n [10x),] = was born [60).] = { ang bang Now Front column of MB) Griversity ( 12 coloukate ) Now, elements of third column of AB [lee), ) = Ž his

Now in brinen, First and Third colnen af gr. we somes son lai, = bis For j = 1, 2n Using This weged, " LLAR), J = 2 azi ba, 1 [[AB), 3] = [(AB), ) Similarla, [(4 0),3] = [(AB),1] Hence, First Same for j=1,2-n and Third column are af 'AB" Part (0) > If The Rows 1" and "3" of "b" are Rows 1 and is of AB. some.. ما دم FALSE ' counter examples

Now, 2 AR= 7 2 31 [2 3 2 4 8 12 u56 oo o = 1 10 20 1789 L 12 3 116 32 48 ol le clearly Rows "," and wgn of are Not some AB

Post (D) (AB) = A BR Answer: FALSE counter Encaple Let A = [68] B= [&: Now, AP= 103/163) = [62] 6=L97 [:1=1937 | An, A4 = 10 + 7 Hence, (AB) = { 1810) *f= [:$][120) = .7 Hence, (ABI & À 82