Question

Question 1 15 pts True or Fate parti s problems, ponts each 1. each como Ahas a pivot, then has only one solution Select (2) de() detA 3) det(A + B) Because A - B = (A - B)(A + B). Select And A+Tare both invertible, then (A+1) - A' + I. Select Because (Null(A)) Col(AT), 50 (Null(A))+ + Col(A). Select 151 17 Ais similar to B. then + 24 + 5/ is equal to B +239 + 51. Select

Answer 1

%. b = Solution • (1) Let A= [ : has prest but Aze=5 hos no solution. Am: Faloe then each column of A 2 3

2 2 •(3) We haree (A-2)(A+B)= AA +AB-BA-BB7A-B; Since AB+BÀ m genoial (matrix multiplication 13 NOT :def[4? B2)+det(A-B) det (A+B) » general. commutatise m general) .. Am: False (3) we find A+1)(4+1)- : (+1)!+ A1+1 And: Faloel = AA HAI+IA +I =A+A++I +1

(4) If we identity matrine 2 we choose Aan the 2x2 then Null(A)= {o}: -- (Nun () ALSO, Col (AT) = 1R, Col(A)=IR? :: Col(A)= (Null(4 IR is Am: False

• (5) If A is similar to B. then we com write A - PBP-I det (P) 70 :: A?= (PBP-"}(PBP-1) = PB(P!P)82-1- PB?p- A°= PB?: Now, A2+ 24351 = PBP-442.PB?p=1+5PIP-1 > A+2435T IS SIMILAR TO B3+ 28751, NOT EQUAL. = P(B3+2B351) -1 and similarly, P- 3 Aros: Falbe

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