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sample test 2 with solns

sample test 2 with solns - MATH205O F08 V2 Name Page 1 of...

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Unformatted text preview: MATH205O F08 V2 Name: Page 1 of 6 (1) (6 marks) Solve the linear systems associated with the following matrices: Ki 1‘. 5‘3 x\. X1. X3 34: 12—3 @50—12 (ii) 006311 00000 X +7.): =—l : __ 7' } X3*XQ=\ ') x;"\’¥‘| \ k SO X1: ' \‘2X3 :3 so- >‘ 1’:le 4*?“ xn‘SxL — Ku =L=2 x‘: whim. 5°- (2) (5 marks) Find conditions on b1,b2,b3 such that the following system is consistent: $1 __ 33.72 —$3 2 b1 25’31 “ 7332 — 3333 2 b2 23-71 __ 53.72 — $3 2 b3 ‘ 3 ‘l '0] 1 '7 ’3 51. 2 S "'l [93 ‘o \ “s -\ I .. , Q Elk—29‘ 0 -\ —2\o.+b,, a3"v~3'2gl 0 " \ -z\o|+\03 l '5 *\ l0: ———9 O | —\ -—2‘o“\\n _ _ . Q3293+Q1 o o o 409‘ +51} (93 system LS couswrem’ “‘4 r“— x.+2kx,_=7. \ 2k 7. R\ Zk z‘\ 2x.\-‘r ‘LLX, - Mk '2. ‘4'” km [um-195‘ o [cl-uh k . . ' 06 SOUR \c w—Rk 140 CAL ‘Lioﬁdul us can 66"! A 6WD LEAMLK» adE, 269‘ /°”‘°‘ W 00w) ' . iv \L'—0 men 9.} qus [o o c] an A cw: WMAWE & w-ww sows k: [LN-l THOJ Q, 25mg [0 o q] 651 m SOLUHB». Page 2 of 6 Name: MATH2050 F08 V2 a d b h u 0 y e r u S e k a (mxw .mﬂ mm mm] god 0010 .m.m W1 1021 10_ _ dA ft0200q|\lJnll|.|Jalll|. mm 0001 000l- OolO HWlOZnﬂOOLQ cote oaO\ 06 un O em_|_o..: ounce o 0 mg __ :1 mm A \||\l I es OO\o OOIo Goal hn dﬁ Mm 020.. o\oo oxoo e \mu/wl ‘07....v ‘000 \000 kW ..I\|||_r|\.\|l\r||\l. r0 \D.. u. ar : an ﬂy in. mm wUJ nmm ,f 50 _ z 5.4 Q (y \\ 0.22 ) A MATH205O F08 V2 Name: Page 3 of 6 (4) (5 marks) For which values of k does A fail to be invertible? (5) (4 marks) Solve for the matrix B in the following matrix identity (hJShOW all 1J0 'T steps, and mind the order of Operations): 1 —2 T —1 _ 1A2; ”News we dues ‘ '3 z ZBT-kiz = ﬁ— [ ] '7. \ Sugxakct I1 Fun Mu SW65 T "q 7. 1% = '1 0 MOUTH-’0! 15‘! i W“ 55°“ X «we 11mm w sides 1 MATH205O F08 V2 Name: Page 4 of 6 (6) (2 marks) Find (AB)42, that is, the fourth row, second column entry of the matrix product AB Where 1 5 0—3 1 g? 831 A— 2 0 3 1 0 B— 2 3 1 5 — 2—2—40—5’ _ 30 2 3 (4104—4; l1—234l —§ (A$)=[~H\ou—ﬂ i 220+310+owg ‘0, a / "L (7) (4 marks) Knowing that ‘ det(A )—1((— and ‘A— 1— _det1( )adj(A (14);)! ﬁnd the entry in row 3 column 2 of the 1nverse i e 1)32, for the following matrix MATH205O F08 v2 Name: Page 5 of 6 (8) (5 marks) True / False. Indicate Whether the following statements are always True or sometimes False. (a) Every linear system has the trivial solution True 3’ False (b) If A, B are square matrices of same size, then (A7L B)2 = A2 + 2AB7L B2 True 3/ False (c) Every singular matrix Has can be expressed as a product of elementary matrices True j False (d) If B is obtained from A by performing an elementary row operation, then B is invertible if and only if A is 2’ True False (e) If the linear system Ax :@has a unique solution then the matrix A is invertible if \K foamy 5 [LAD M51100 new 11's 1106' True False (f) A homogeneous system of 3 equations in 5 unknowns must have inﬁnitely many solutions 3/ True False (g) The permutation (4, 5, 3, 1, 2) is even “'5‘" L"° 3/ True False (h) If A and B are matrices of the same size, then det(A + B) = det(A) + det(B) True Q, False (i) If , R”, an V” t 11 V True False (j) If ,V, 77’ ui V w,then True False MATH205O F08 V2 Name: Page 6 of 6 (9) (4 marks) (a) Show that if A and B are s n1n1etric, then so is A — B. so (NED: JAKE: 7 (saw (IX—(5)1: A-Pm’ (A- by = A19; (EA—G: A2\$ ME WHMWIG- (b) Suppose that A and B are invertible matrices, and A, B, C, D are of same size. Show that if A‘lBD = C, then D = B‘lAC'. A49”) = C =) Ra“ (51) = 9:;‘AC =9 A “4% ‘b = AC =) ID: PJ‘AC, =)(“Db=AC ...
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