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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
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5 lb
: L‘ 7344‘ __5 '2 : ,Ll(‘%+k.‘\) .. S(2—\‘§>
lb 6‘
= ~4(u)— 5+6)
:*¥>&+GS
3 5—84
<b>A= 83 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, moltH3 ) (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 willw 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 (38 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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 Fall '07
 Wilson
 Linear Algebra, 0°, Invertible matrix

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