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Unformatted text preview: Math 304 Name: K E 10/4/2010 Section: Your Score a In order to get credit, you must show all work.
0 Read all problems ﬁrst. Try working on them from easiest to hardest.
o No calculators of any kind! No cell phones! . . 1 0 ——1 —1 1 0
1. Con81der the matrices A — (2 0 1 > and B w ( 1 1 4). For each of the following operations, either do the indicated calculations or explain Why
it is not deﬁned. ‘ (a) 3A+2B (b) AB (c) BA
(d) ABT (e) ABT—(B.AT)T (f) (AAT)2 ‘ ~~3©"3 'Z.2_o_ k+2___3
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@DAgdﬁNY:A§?QOW§:A§FA «y; —0 cl 6>®Nlilliﬂlgt 2. Given the following system of linear equations, Where t is some constant:
x1+2w2+m3+x4+m5=1
x3 — 934 + m5 = —1
x1 + 21102 +’ 01133 + 2564 + 0335 = t.
(a) Determine the augmented matrix of the system and identify the coefficient matrix. Then bring the associated augmented matrix of the system into row echelon form,
indicating all elementary row operations. . l 2— l l l Qa£~r\ll \l vi?qu
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00630th (b) Give an explicit value for t such that the system is inconsistent. Then determine a
value if for which the system is consistent. (SasJCew/x is consisiiavﬂi' ‘uﬂ owxcl mild Cl: t’llo.
So “ml is Cans/{smut 0M6 ¥or til.
It is Magus/Eleni %rwmplelrggrt10, (c) Using the value for t which yields a consistent system, determine the reduced row
echelon form for the system. Indicate all elementary row operations. ‘lZ\\\\r..>u_g\202.02_
o0\\\\A> oo\~\\*\
000000 0063000 (d) Identify the free and the basic variables of the system. Then determine the general
solution of the s stem and check your result. (PW “W (€33 X1, XL] zyf ‘/ BUS/k, VatVfableS; XUXS , sanction: chat X‘: 7.”27<7." Z—Xq (1*2‘ﬂz’2/5Lh‘t Zfzfel “N‘s” XSB txuﬂ 7(5‘ 1 l \/ 7%“:qu «bx 'Xs CH “WEB” Y” “5‘24 V
_ ~— Ll ~ — _ h— it: XL, (2 2x1 ng5+sz+ 2m!  L \/ 13 110 —1
3.LetA=(2 7) andB——<2 0 2 1). (a) Find A‘1 and check your result.
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W M“: ('2: [EH ‘6] v 0 (b) Use your work from part (a) to express A—1 and then A as a product of elementary
matrices. Then check your result. it “alarm? ii: = T . 36 ,_.. L’K‘J ~ [Chick
A; [HEX MA ﬁll; ii: [L :1
«and» (c) Solve the matrix equation AX = B by using A‘1 from part (a). (You must use A‘l,
not any other method.) ,4 ,‘1'3 \lO'l__ [tiné “‘0
Y'AB‘ ’1l3[zo?~l‘[ A :
i :1 x1+m2+w3
4. LetF 2 = $2+m4 .
.171 +373 "—174 (a) What is the domain of F and what is the codomain? 3
(DOMOCLVI ‘. KL. Cﬁéﬁmcu‘th IR l (b) Determine F(ei), z' = 1, 2, 3,4, for the standard basis {61, e2,e3,e4} of R4 written as
column vectors. i [$330]: 1t 0‘ \+O‘O L C) o
\ "i (c) Using (b), exhibit two vectors v,w E R4 with F(v) = F(w) but v yé w. Is F
one—to—one? Explain. 1 (:0? V3€4 MA I 01'
33 gag—CM‘HOV‘) F (5 mi‘ Owl~+1>~oqu\ (d) Using (b), write down the standard matrix A of F such that F(x) 2 Ax. WHO
0’02
(OIl (e) Determine the rank of A and show F is not onto. .\ \ s 0 WET \\ IO (ganja \ \_ \O anA22<$ oi©(5‘Oox—~—?O@OC\D gomnkvéeies: lO‘—‘ “ "i 000 HAMHULW‘ (Fe
O \O wasmsi: is MJ!‘ Cid—0‘
(f) Exhibit a vector w E R3 which is not in the image of F.
r . O a
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Ol©ibILg>nOlOtb 3>@\O\bb ‘ ‘ C0 O O O O Q’OW
$6 Ckng VQdYDY M‘H‘ Q'OUrLﬁEO noi‘ [:2 [k W
mete 0i F For were [riﬂe EMF. . Circle “True” or “False” for each of the following problems. Circle “True” only if the statement is always true. No explanation necessary.
’34.. (a) n m X n matrix has m columns and 71 rows. v A,. (b)‘ If a 6 X 8 matrix A has rank 4, then any row echelon form has two zero
rows. (0 Suppose that A is a 5 X 6 matrix. Then Ax = 0 has inﬁnitely many solutions. as V
(d) Let A be an m x m matrix. Then Ax = b has a solution x for any b E Rm. '
' AV
(6 ‘ The rank of a 12 x 6 matrix is less than or equal to 6.
(f) , he row echelon form of an 8 X 5 matrix contains at least 5 zero rows. (g) The matrix 3) i (2) is in reduced row echelon form. A
118 r . 2 2 1 1 . .
(h ‘ The matrlx [O 0 3 0] is in row echelon form. (1 Every elementary matrix is nonsingular,
0 Let AT  B be deﬁned. Then (ATB)T : BT . A. ...
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