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Question 15: Do the vectors below form a basis for R3? If so, explain. If not remove as many vectors as you need to form a basis and show that the resulting set of vectors form a basis for R3. C1 = -- () -- () -- () -- (1) Question 16: Carefully consider if equalities below valid for matrices. For each equality state if there is a restriction on dimensions under which they are valid or are there any other properties of matrices for which each might be valid, if ever? (a) 2(A - B) = 2A - 2B (b) (A - B)2 = A? - 2AB + B2 (c) (A - B)(A + B) = A - B b "C)and ) C Question 17: Find conditions on a, b, c, d such that A = commutes with both Question 18: If Aand Bare symmetricn x n matrices prove that AB is symmetric or give an example that A Bneed not be symmetric Question 19: 2+31 (a) Express in the form a + ib (b) Calculate: (21)

Answer 1

D (A–B)2 – (A-B) (A-B) A - BA - AB+B² JP A,B are non-commutative, then ABEBA. So, the given equality is invalid in general -B) (A+B) A²-BATAB-B² If AB EBA, then the given equality is not valid It 12 A(66)=69)A (21)69)-3) (ao) - (0 ) So, @ b=0, e=0 Also, A(60-69)A >(21)60) 40B C Hence. The conditions are 6=e=0 Tad can be any real

187 AT A,BT = B Now, (AB)T - BAT-BA #AB in general Take A= 132 62) B 2 - Then AT=A, BT = B AB - (62)(23):G4) = - AB [ 4 symmetrie. 19.>@ -2732 = (AB)T 27 So, AB is not (-2+3 i) (93-4 i 3+4i (3148) (3-42) must -6+8 it dit 12 6+1fi 9+16 25 25th 25 (6 (28) t = 24 84 - 16.1-16 +2 17

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