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20110426074956451

# 20110426074956451 - Cormch Esluﬂ 6M3 J‘ Math 2J Spring...

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Unformatted text preview: Cormch Esluﬂ 6M3 J‘ Math 2J, Spring 2011 Practice problems for the midterm 1. Determine whether the statement is true or false. (a) Overdetermined systems of equations always have no solution. (1)) Row equivalent matrices have equal determinants (e If zero is not an eigenvalue of the matrix A, then A is invertible. 2. (a) Give an example of an underdetermined system. (1)) Give an example of an overdeterinined system. (0 Give an example of augmented matrix for a system of 3 equations in 4 unknowns with inﬁnitely many solutions. ((1) Is there any system of 3 equations in 4 unknowns with a unique solution? If so, write down an example. 3. (a) If det(A) = 4 and det(B) = 5, what is det(AB)? What is det(A‘1)? (b) If B is a 5 X 5 matrix with determinant 3, ﬁnd the determinant of the matrix —ZB. (c) If C is a G x 6 matrix with determinant 5, ﬁnd the determinant of the matrix obtained by C by first switching the second and fourth row, and then dividing the third column by 2. 4. Let A be a 5 x 5 matrix such that A = —A". Compute det(A). 5. Solve the following system of equations. ( Use any method you like.) 331+ 21132 '— 4333 = 0, 3(31 + \$2 + Ll:ng = 21, 2531 + 41132 — 6313 = 1 0 2 —3 . ( Use any method you 1 3 6. Compute the inverse of the matrix A = HHH like.) 7. Calculate the following determinant. det 8. Find the eigenvalues and eigenveetors of the matrix A : ( 1 [1) Is A diagﬁ onalizable? ONO 1 9. (a) Calculate the eigenvalues of the matrix A = ( 0 1 1 0 . 1 (b) Find all eigenvectors of A. (c) Find an invertible matrix S and a diagonal matrix D such that S‘IAS z: D. ((1) Compute e". K1®E rmM brqu sokvk‘mn Q-i‘zo\ '\ 2- MA ohbj \’(‘\ ‘3 ‘mvu‘x‘;b\e ‘ ova/4 Q— A! '“M’V‘ Now 1!; o *3 “°* °"‘" 6W“ ‘A *5 mo 1ter «2»ch 49' ski A19 ; 019 r— 5r O\ “on o “6 T” L m a when M3" A“ 6 \LM Vt a mu \qamo. MOW'LMO 34\$ A ‘3 (IV rat/W J“ New \ a0 oqwkok/rm‘meti . 7.x. X : C23 ‘39 ¥ Z a @ 7,)“-le xq'VX’Lflo [Ah Dundef mama SAW‘I’ICWU 0"" “or We Amgwm {5 Lg 9x00 E‘IU‘M—Y ‘MQ—(w'll'f-IC SOL/xsr‘tOh- @//—W AC it We. have. :_A :3 Ad (Ah JJQA) 5 :44) rig/HAW: - er—AMM :7 ideﬂmzo :prﬁkMzd ) @ ‘ 1 ' Lt 0 AAA L—NKR‘ \FORL ( 3 1 q 2[) MAM—Ml. iroRz 2 H -6 l \ 1 _H O o 4‘3 ‘6 \21- o 0 +9— 2- ><\»rlxq__‘4)kg:o “’"[email protected] H“) 6X2*l6K3 /? X2: _\ \ ‘ O \ O 0 AC” (.“1KRI +5121 \ 1 .39 O \ O \ \ 5 O O \ AcH \—\\AR‘*GRQ o \ -3 -\ l o H 0 c9 3 J O \ \ k C ‘ O O Aux—mm Jro 12‘ O \ o ’1 \ M O o 3 —\ O \ ‘ O O 3 4 “ Wasn‘ij R; \W} C) l O «7. \ \ fr): 6 O 3 A O \ K O O ’3 --\ -—‘\ O k 0 .32. \ \ \ R O C.) \ "/3 0 K3 ) 3 "\ F \ 1:) A” ’3. ,7. \ ( 1/ 0 V3 ...
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