208midterm1solutions - 1(18 points Compute the determinants...

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Unformatted text preview: 1. (18 points) Compute the determinants of the following matrices. 2 3 ——4 (a) A: O 5 5 1 6 Z 3 ‘L‘ .LH 34" U2 2’4 Hosié‘)‘1\c*e‘)ose 5 lb : -L‘ 7344‘ __5 '2 : ,Ll(‘%+k.‘\) .. S(2-—\‘§> lb 6‘ = ~4(u)— 5+6) :*¥>&+GS 3 5—84 <b>A= 8-3 12 0 0 02 325:3 ‘4‘ \>‘ 3 s vskl OOH—g 5-. - Vedas LC) =G‘\C33(~ 0 0‘0‘2 00024 [0 . _ 1 h _ £131 _ 0 '__ 2 2. (16polnts)LetA—[4 8],x—[ d],0—[0],andb—|:k] (a) For what value(s) of h and k does the system Ax = b have exactly one solution? ‘ k‘t‘l l L\ 7— [Li 1 \< (39% lax] Ax:b M5 maul/l1 onesolulfi‘omll' <5’thio } K’cé : O‘Y‘Y‘rt (b) For what value(s) of h and I; does the system Ax = b have no solution? B—Llhzo k'&¢0 (e) For What value(s) of h and I; does the system Ax = b have infinitely many solutions? 3*L—lhso) ls'&=0 =3?h;2, molt-H3 ) (d) For What value(s) of h is the system Ax = 0 consistent? any r—ml‘“: 1 34 (16 points) Consider the following matrix: A : [ 1 3 (a) Find the inverse of the matrix (if it exitsts). (b) Use the inverse of the matrix to solve the system at +52 : 2 at +y 2: 1 390 +2y +62 = 0 X A b do —°\ 5 l -L\ —\ «7. \ O ~'~\ MW; r) .. 3 . (16 points) Consider the matrix A = [ 1 3 (a) Express the matrix A as a product of elementary matrices. 23] 9169?; €\¢ [0’] ) g‘h‘: 1 ’5 10 \ 3 d x, [o —1 7 L 0] z ’5] La‘le 6 ' [’2.\ 3 E2 ‘2 \ (b) Does the system of equations represented by the matrix equation Ax = 0 have a unique solution where 0 = [ 3 ], x 2 [ :1 ], and A is the matrix given above? If so, then 2 What is the solution? Ye; \ smu A Cow» la will-w N Avomt 9!; eumnkmy wl-(CJJ AX‘VD Ma) Ukk‘l‘~‘% «\fiw‘q 8\VV\\VV\ X‘) : 1 _ 5. (12 points) Let A 2 l identity matrix. \“3 L H 2AL=§ [ \ aA2_A: ~10 *b]_ \-3] : ~‘\ Li -\Z [2, 0 Z &A2~A+%: : [:u 231%? 0] t z ~r2, o \ : —2 —3 [2. - «w ?CA\ 2' 2A1 «A A 3i; ~ oli \ -3 20 1—3 9 0 .4 If" \t*9 L’rU I3+O ~bxo 1 . Find p(A) if 10(1?) 2 29:2 — it? + 31'2, where 13 is the 2 x 2 6. (12 points) Solve the following system of equations. Determine if the system has a unique solution, no solution, or infinitely many solutions. If the system has infinitely many solutions then express the solution in parametric form. 2131 +3272 +4133: 7 3171 +9172 +7123: 6 ‘3 9‘ ’3} N \'?>"l [3% _-l\(0] [a (3-8 fil‘S‘ " [‘ °’ W] ” C‘ ’3’ “T? OO\ 99 0 0| 3 7. (10 points) Determine whether or not the following statements are true or false. Justify your answer by giving a brief explanation or providing an example of Why the statement is false. (a) If A and B are any two square matrices of the same size then AB=BA. (b) If A and B are invertible matrices the so is A + B. (c) If A and B are invertible matrices then so is AB. ((11) If A is non—singular then A is row—equivalent to the identity matrix In. (e) If A is a singular matrix and AB 2 AC for any two matrices B and C (of the appropriate size) then B = C. A g B Ar7, _ 33 M3 1? a» as men: Bil ) 0.351834%} *5 4 A3 ;nVev\;l.l<, \Wl’A.+3 lAMO+ A ,Eo : of) 5) am gable—L 0° 8. Given that A is a 2 x 2 matrix and (2A)‘1 = [ I on” "v twmy 61A “w x Lg » W”: \ f 2 w ‘ g t w ; / i 114%» “in; MM“ 3 M; "“U; .AL if“ H Xi ii v 5‘ g H", i M; “i "3 i h ' 12 34 find AT. ...
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