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Unformatted text preview: THE UNIVERSITY OF WESTERN ONTARIO
Department of Applied Mathematics London Ontario
Applied Mathematics 025b
Second Tutorial Test w February 21, 2008
60 min Name: —_ Section 2 Closed book. Only simple calculators are allowed. Print your name above
and student number on the page 2. Justify answers by showing sufﬁcient
work to get the full marks. Solve each problem in space provided for that
speciﬁc problem. Use margins or back of each sheet to do your rough work. Instructor — Dr. N. Kiriushcheva PART A. Mark the correct answer. [2] 1.1. Given two rectangular matrices A and B, (a) it is always true that AB = BA;
(b) it 's never true that AB 2 BA;
7, (c) ' is only true that AB = BA when A and B are square matrices;
@ it. is sometimes true that AB : BA, but not always;
. (e) in the special case that A and B are nonsingular. then it is true that AB = BA. [4] 1.2. TrueFalse Questions: (a)? product of two elementary matrices is an elementary matrix.
11 False
3 e (b) if is a singular n x a matrix, then the reduced row—echelon form of A has at least.
On row of zeros. False (c) A product, A X adj[A), is a diagonal matrix only if A is a diagonal matrix. True als (d) For any gr x 71 matrix det(I + A) = 1 + det(A). my AN! 025k) Test 2  February 21, 2008 1’ Student #: FOR INSTRUCTOR’S USE ONLY III R
9, V _ f? O / [6] 2.1. Suppose that A is 4 x 5, B is 4 x 5, C is 5 x 2, D is x 2, E155 x 4. Which of the
W fellow" mg are not deﬁned. if @4891; (b) )AC+D @/AE+B/ @/J93+B (e) E(A+B); (f) MAC); @JEY/A (h) ).(AT+E)D Note: there might be multiple correct answers. /,
)3 then A is / 6/(32 33 —3 —1
5 2 7/ (at? 3) (b)(f”1§2), (d) (a: 3%), (e) none of the above. 2.2. 19 (5319)"1 = ( AIN'I 025b Test 2 — February 21, 2008 3 [8] 3.1. Given that det[A) = 15, det(B) 2 5., det(C) = 3, the value of det (fl—183.0) is
;: an LA”) . an C IN). MW) e “1...... . ow 5 auto)
(b)225; dum ( j r. (C) 25; __"l_ ' ._ ' 3
{5 /
© 1; \/ (e) cannot be determined from the information. (a) 15; r l {I i333 WV“
.r ' . ' I? I 
3.2. What 15.13110 value of the determinant det 0 Q 7 0 ’ {ﬂ} _ O
0 0 0 2
.0; a Q ‘P
G“
(b) 2; ' 0 o 2} .
 : ’2 G '
(0)2. a ”1k \1 3 1
(d) 14;
? 11'1“ch D
(e) 14.
2 1 —1 3 A a *3
3.3. LetA—(_l 3),andB—(2 1). “64:42?”
Find an elementary matrix B such that B = E . “ 1 *3? E: «We. 2), eke a, (d) E : (3 [1]), (e) none of the above. 3.4. Find a and 33 such that det (2' 3) = 0 : det (G 4). (D an K 35.73] :. o l
Cato—bio
Quzb ' ARI 0251) Test 2 — February 21, 2008 51 PART B. Show all steps of your calculations. [10] 4/1 Solve the following system of linear equations Ax:b. where 1 1 1 0
A: 2 —1 ql and be: 1
0 1 2 2 My ﬁnding the inverse ﬁrst (Alf) —> (III1‘1), and then solving x = A‘lb. Musing Cramer’s rule. whiz, It you) 1\\'=IL’>D" 11.1100
‘ (l) __{ _ 0H") allRam? 31340 R113 0 33.43 0
'3. l '
0 oo  1
011 ool 0121001 AM O25b Test 2 — February 21, 2008 5 [10] 5/ Determine the. value of r1. for which the following system has no solution, exactly one solution, or inﬁnitely many solutions. 131 +332 + 7533 = —7 : NAIRd ”AM/MK I, 23:1 + 351:2 +1733 : —16 331+ 2332 + (a2 +1_);.::3 = 3a m»; M "4 “'4'
I?) _
.0?» ‘c‘ O AM 025?) L3} 31 [10} ﬁ/Suppose that A: ( q Find: «idea—3A) abc ghi Test 2 — February 21, 2008 d e If) and det(A) = 4. {(55 dam1) Wdetmf‘) c b —a
(ea/deMBA‘UT {Q/det (f 'e —d) {5 ART 0251) [10] 7./"C0nsider the matrices H Compute the followin (Acqaq: M‘BGI— IAC.
Mme) MAC (01A )1, 1 1 g: Test 2  February 21, 2008 J: easy 9.2 AC=[ ]['+IJ [elrz *3]
.—[L 1m, 1 (LA [iv—l]
(laCA A —3 o
1:: l— ['53: 3] (i ‘5) r=( m—UFBB'H (ff (2A0)T — OTAT. [2" ”ﬂ
: 1 09—] 10
01 J ...
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