Question

a с е f such that all the entries are b Given that A = d g h nonzero and |A| = 5. (a) Evaluate the following determinants. (1) |2g – a3 2h – a² b 2i – a²c| d f b е ܢ a с a с b с e f (ii) d f 0 7 4 0 g h i (b) What is the volume of the parallelepiped generated by (2d, 2a, 2g), ū = (e, b, h), ū = (f,c,i). U =

Answer 1

Given that A= the call such that a b c d e f 9 h hi entries ars non ZORO and lAl=5 @).) 27-28 2h-ab ziarc + d e b C a 29-98 +48 26-ot tatb zinate 5 toc R, Rita Rz f R a b C 27 2h 21 - C f R a b c = 2 ot h [Taking common 2 in RI R1] d C f b a C

= 1 2 (-1) 4.w [interchanging Ri and Re] of ooo h b C de = -2* (-1) f [intorchanging R₂ and R3] a b с y h i a b С = 2 (-1) [interchanging R, and RL] d C + g n = dat -2x 1 A1 -2x5 Hence = -10 27-a3 2n-arb zicare f *=E 10 d C a b C

(1) C df 7 b c f e 40 ore 4 h 0*7 b c a + 4 expanding with thired row d f f I to def gh g -7 11 a b de In с 7 + 4x0 [ since two calumas C, and I are identical] 5-71A) +0 --7x5 = - 35 Hence b c f - 35 a с d f 07 gih 0 (6) Given ū ( 20, 24, 25), ū- ( 2d, 24, 27), D: <c, bah) W=(fcit The volume of the parallelepiped is [ü vw 2d 2 a 29 e b n f C = 2 d a q Dust b h с de t = 2 [. dettet | A+) = 141] ab In = 26-01 a ь с det [: interchanging R, and Re] g h i always 26-12 * 5. =-10 volume is positive Hence volume of the parallelepiped is 107 cubic unit