Question

Please use matrix method

0 0 1 2. Let A = 0 TO 0 0 1 [ 1 1 2 21 jo 1 0 0 1. BT = 2 2 3 3 0 0 1 1 (1 000 (1) Calculate BT AB. (20 points) (2) Calculate the rank and nullity of BT AB. (10 points)

Answer 1

# solution. Given that the matrise r. 1227 Too 1 o. B-12 2 3 311 Loo 1 1 tooo L 11 оор L1 o daxq then (BT) = B = I 2 1 2 3 O o 1 2 1 ww ve Pro 24x3 so , oop o o od 14x9 [11227 2 2 3 3 1 0 To 0 1 1 "-3xla (2 2 1 3 3 2 2 BTA = اکر d 3x4 2 1 1ro . (OTA)B = BTAB = Tp wn rooo opo ooor 2x

and BTA)B=5 1 3 1 2 2 17 11 2 07 3 2 2 1 1 2 a 1 1 0 0 3 2 xel 2 3 1 days wwnn 2+2 +2+2 e +4+3+3 0 tot 1+1 BTAB = 3+3+9+e 6+6+6+6 oto+2+2 1+1 toto otototo to 2+2toto le 2 BTAB = 24 1 2 4 0 Answer Let P =BTAB then 58 10 27 Eli 20 e 1 2 4 0 13x3 determiant of pis Ill = 8 (0-16) +14 (8-0) +2(56–48) 1752 ela -12 6 + 112 + 16 (11ral Leluto Pl= 0 So IPI = 0 Rank (L) <3 le pr8 24 27 (R2R, R2 de P=2 40 & ft

P- 7 1 8 2 10 2 & o ( R3 R3-R2 Rank (L) = Rank (BT AB) = 2 By using the Rank - Nullity theorem is Rank (el+ Nullity (R)= order of the matris (2) Rank (L) + Nullity (2)=3 > 2 + Nullity (21=3 Nullity (2)=3-2 Nullity (2)=1] BTAB=P) le. Nullity (BTAB) = 1