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# midterm1_green - z SoLuuou's Gﬂé‘E‘M e-xmA SECTION...

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Unformatted text preview: z SoLuuou's Gﬂé‘E‘M e-xmA ) SECTION: A (10:30—11:20); B (113042.20) NAME: MATH 214241, MATRIX ALGEBRA Spring 2011 Midterm l _ Total: 100 points a You must show ALL work for full credit. 0 If you have any questions during the exam, please raise your hand. 1. (20 pts) a) (5 pts) Construct the 2 x 2 matrix having 09' = (4)1}; A _ i: 1. U2 .1 .L b) (5 pts) Show that (A2)T = (ATP. 0%)"- .—. (A+)T = ATAT = (#01 c) (5 pts) True or false: if rows 1 and 3 of B are the same, so are rows 1 and 3 of AB. Justify your answer. @' row (of 1H5 :1 Crow {WP-Pr)? d) (5 pts) A system of equations with inﬁnitely many solutions is said to be ...99-.r.~i§i§r§m ........................ and ........ smwz, ............................ ' 2. (18 pts) (a) (8 pts) HA is symmetric, prove that (BA—1)T(A*‘BT)‘1 = I. (Simplify the product by using the properties of inverses and transposes.) ' (BA“)T(A-**3T)"‘ = CA“? N MYTH)“ I. : AT)“ (A4)“ A as; . :M 0:“ 3tsﬂ"* '3- T ..— . b) (5 pts) Prove that the transpose of an upper triangular matrix is lower triangular. L91 t"? @1an MW Mmﬁiﬁm maid): ; a“? aolg><¥ 5mm. Q30 =0 Myer Hm. (0w mcﬁtx is £151 4km +5“- L-UQLUmn ‘lmﬁJl/‘K' AT is ROW ﬁmww c) (5 pts) True or faise: If AB and BA are deﬁned, then AB and your answer. frag Wat =) A: mm, 8.4“qu BPr'OKljF‘ruﬁf. «1% I(‘ﬂkp>(tmxﬂ) a) Wt”?- So- Anmzxn ‘. anxm. BA are square. Justify M ﬁg): imx‘m ngum) TM): 1 run CSiuQﬂ)‘ 3. (25 pts) Use the LU factorization ofA (or of PA, if necessary) to solve the system: x + 3y m z = —1 2x + 5y + z = 4 2x + 7y — 42 = —6 Hint: Obtain the two triangular systems, as done in class and on the homework. If you need to do a row exchange iorA. don't forget to change the rows in b as well! Use the back of the page if needed. l 3 -l x *l a 5 1 t4 ' H 2. 1 ‘ll 53 *6 —.—'3 A i? 3:: l 3 ~\ I 3 ~l -—_.___; ___‘_.x) 4&4“: . 0 l *1 o 0 l l O 0 L 2 i 0 Z "I I a, l O 0 Cl —l C :4 LE 3‘0 =) ‘ ,C.) 7 _ i Z C2! ‘— H :) g[_{)l—(Z=q s) C2,: C l ‘l 2 C5 “'6: -1 “ff-l (aw/g a) C352, DIE) : 8 _} I ,5 ‘1' x? : “E % :2. O ‘1 3 ‘1 g a) _\4+ﬁ/:ng :Dljzo o o I e} z 4. (20 pts) Find the inverse of the following matrix by using the Gauss-Jordan method. Clearly indicate at each step the row operations used. You must show all the steps. Your inverse should not contain any decimal values. ml 2 —1 A: 2—1 0 0—1 2 s1 2. —\ 1 o —l 2 “l [ l o O O l O __.__...____> W l 2 l O (L —‘ O ﬂat-Preﬁx o 3 2 O i‘ 2, 0 O i 0 _‘ Z O O 1 ___;; ~1 2 “I g l O o (-4 ‘2. -l l o o ,__, , 3: MW!) E! L 3539. O m i 2' 1 O Q E 31214423 o .. i 2 O 0 I 0 5 ‘1‘? l o 0 0 LI 2 E 5 { sl 0 5 l O 7. #1 o 0 1‘7")“; ‘3/‘1 *n‘f ._._? l . It ”A? .. -L; wilt, “1/ QQ‘l-x-(Eg [0 h‘ 2- O O ] “923+Q4 O t L?) ‘3 It; (1 Rs/L; O O i Z/q lit,k 31L] bmr3+ﬁb t) 0 l 1 Elle ‘/(l 3h, l o o lzlt‘ 3m 37!? :2: o i 0 Lm} 119 in, o o l “‘1 4f? 51% l A" . l [‘2‘ 7’ 5 H LL; 1 Z 1 I 5 5. (17_ pus) LetA be a 3 x 3 symmetric matrix with pivots —1,1,2 (in the standard order). The elimination factors lg- needed to putA in upper triangular form Uare, in order, —1,0,3. By using this information, ﬁnd the matrix A. Hint: Use thatA is symmetric and the LDU factorization of such a matrix. 1' o o a; L: _ D t 0 o .7 —\ \ 0 r o_ i o 0 3 i o 0 Z Siam Wis 53mmm, U=LT = i " 0 O i 3 o 0 | A:LDLT :- ‘ 0 0 hi ‘ O ‘I i ‘0 b i 3 ...
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midterm1_green - z SoLuuou's Gﬂé‘E‘M e-xmA SECTION...

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