Math235MidtermExamSolutionsF08

# Math235MidtermExamSolutionsF08 - Math 235 Midterm Exam Fall...

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Unformatted text preview: Math 235 Midterm Exam, Fall 2008 Page 2 of 10 Name: 1. Suppose that A and B are n x n matrices over a ﬁeld with detA = 3 and detB = —2. Find: (1) det(2A). 1 h J '3: (ii) det(B‘1). \ — Z '— (iii) det(ASB§). 3 ’L 2) 2 ('05? (iv) det(AT). 3 Math 235 Midterm Exam, Fall 2008 Page 3 of 10 Name: Emu—hm— 2. State Cramer’s rule for the equation Ax: b. ‘ V . A am («A Verﬁkiw an WxaTn‘Y‘ +9.2 (Li/wa goCuotnoa {'22 “=5 1‘s M“! £7 , . " x: @M/é) Mm ﬂz/5/=/4-~ AM we ‘ W) / (ii) Use Cramer’s rule to ﬁnd \$3 in the equation , / Lﬂwéz Mb Math 235 Midterm Exam, Fall 2008 Page 4 of 10 Name: MM 3. Let V be the vector space of upper triangular 2 X 2 complex matrices. Let T : V —+ V be the linear operator whose matrix representation with respect to the ordered basis ' Ml?) 3H3 3H3 Bil} is Find (3 [T]C where letv'S’; [gig an! «Aug, . V 0 \QO 1; PM e‘ w g2“. 0‘0 0“) col l°?Ll:;? "” ow O Math 235 Midterm Exam, Fall 2008 Page 5 of 10 Name: 4 Let V be the real vector space of smooth functions on the real line spanned by the sin and cos functions. Let T : V ——> V be the differential operator T( f) = f’. Find det(T). - V: P7l)G\R gig : g 3W“)le ABM/33% V Math 235 Midterm Exam, Fall 2008 Page 6 of 10 Name: 5. Let T be a linear operator on a vector space V and let W be a subspace of V. (i) What does it mean when we say W is T—invariant? I'll Mans JC‘lmac’l‘ «Cw my wéw 9 'T(w)€\l\/. (ii) Prove that the subspace Null (T2) is T—invariant. Lgt. (A: 6- Nu/QQCTl) Eelmlullwng given. ‘3 —— Tlnen \ < \ U“) :2 T (T (T (0:13)) , A Q lab Mg‘hilqmq CV31 Emportllm o; 2 T (szu) WW”) Tb‘é DWVQS Hwb MM (TI) (3 thVﬂ/m‘qvxﬁ Math 235 Midterm Exam, Fall 2008 Page 7 of 10 Name: “W 6. Consider the real matrix :2. ll CON Hklol—t 0000 Find: (i) Its characteristic polynomial. ->\ I o 7 "2-%‘ .3” JPAKMZOUJ— 2c: 2m 0 XZ‘LL’CR‘ 1&3 C \) O 0 4—3‘” ~_—_ lz-le—m) =— wzfmam- (ii) Its eigenvalues. )ﬂAZ. ’ (iii) A basis for each of its eigenspaces. Math 235 Midterm'Exam, Fall 2008 Page 8 of 10 Name: 7. Let A be an invertible matrix and let A be an eigenvalue of A. (i) Show that /\ 7€ 0. [4/ A4 (lav. k 00+?)t'} -. ‘/ ’hzlh_ (:#(A-61)) if <2 )bmu» 4° 2 m; lambda A» AHA)¢0 m 0.240 , 7’- laae /:0 {0 no!“ a no} 4 79M (10;. Psi/3. 4 ,4 We Iona/null Jam} ﬂfo lthr’TM 75b» ”0 ” /o»4.,/- k a. no} 4 #4 Mar. fol). any am/ ﬂu ” ” 0 4; mal- Q-r etyenva/ug . (ii) Show that 2/\ + % is an eigenvalue of 2A + A4. 3 vd-o & Iii/3 )‘l’ Q (J'm') -I -1 AL» 51.; ,gmhba n, AM- ﬂ-Ay : A u.- xx'g m/ AW: X’_V@CA¢°¢- @) x4140 mm fwmﬂ M 2/43: 2%}: III ﬂaw/2} ﬂan/ﬂ yew 2/71! 154“! : 2 )1“; X’v 794* (a (am/11y: [21+X’)_v VG»? Vi—O M Ant/[4404 l/ In ftwh'Wt/El’ c4 " "/ éﬂiﬁ”) MA! in)». mm (2 Mx'}_ Math 235 Midterm Exam, Fall 2008 Page 9 of 10 Name: 8. For each of the following statements, indicate if the statement is true or false in the box. Brieﬂy justify your answer by explaining why it’s true or by providing a counterexample. (a) Let A and B be n x n matrices. If A can be row reduced to B then detA = detB. (b) For a 4 X 4 matrix [adj], a12a23a42a31 is a term in the expansion of detA along with an ap- propriate sign. 74mg“MA,w/;AM' Wm»; (ﬁom #4 fern-e té/vmn. (4/2 0*9’ ’92) (0) Every vector belonging to an eigenspace of a linear operator T is an eigenvector of T. 7‘12) -.— 2 7k 3%» (fog/3). [.0 (Gal; elﬁwym 74)“ Wth' 5‘0"“. an away Pet/Ur- _———- (d) If A is similar to B then A2 is similar to B2. 3; I'M/V: I] _ A» War fﬂP: 5-} 74c; .;, f}; (2MP: F’A‘P: )3 1 1° A: «flay/)0; 7: .52 (e) If A2 is similar to B2 then A is similar to B. 4» J: [57) M! A’: XL") .04“ ,72; 52:51 A“ A1,»! 5’ 4K J/r-af/m I-IJ' ﬂ’»/ £0: ’10" W»; 174; — —1 . . (f) The real matrices 33] and [ 02 3] are Simllar. 74“ £24. mine; 1m W ham” (‘laaa’)) a”! 2%» ﬂy AWW‘R ...
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Math235MidtermExamSolutionsF08 - Math 235 Midterm Exam Fall...

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