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Unformatted text preview: ,, .iﬁ_—_______..—__—._..u—___________~____________u__. Prof. Dancis MATH 461 Test 1 June 17, 2003 Use the methods of this course. Calculations should be labelled and easily understood. You may use the results of your calculations in one problem to shortcut calculations elsewhere. The
use of a hand calculator is NOT permitted. Use this info to shortcut calculations. ab
cd If M=< ),then M“1 = (16th < d —b),whenever detMaéO. —C a 1. (a) State the deﬁnitons for (i) probability/ stochastic matrix (ii) a steady—state vector. 6
points
(b) What is the connection between the census vector and the fraction vector? 3 points (0) What is the reason or motivation for the deﬁnition of matrix times matrix multiplication. (One equation will do.) 3 points
((1) State two speciﬁc 2 x 2—matrices, A and B, such that AB # BA. 4 points
(e) State the (two) deﬁning equations for inverse matrices. 4 points
2. (a) State verbally what multiplication (on the right) by the ﬁrst standard basis vector, (3))
does to a 2 x 2~matrix? 3 points
(b). State verbally what multiplication by the matrix (3 g), on the left, does to a 2 X 2—
matrix? 4 points
2 0 0
(c). Find the inverse matrix, M‘l, when M = 0 3 0 8 points
0 0 4
3. Prove the Law of Cosines using dot products. 10 points
4 Write this pair of equations in matrix—vector form:
7m + 3y 2 1
a: — y = 7 6 points
Use the formula on the top of the page, to solve this pair of equations. 8 points 5. (a) Given that A,B and C are three unknown invertible 7x7—matrices. State the formulas for (AB)_1 and (ABC).1 4 points
_ ~1 _ Starting—with/ assuming the formula for (AB) 1, prove the formula for (ABC) . 6 pomts (b) Given unspeciﬁed 7 x 7—matrices M,N and P, where P is invertible, and M : P‘lNP.
Show that M4 = P‘1N4P 12 points 6. Find all points (coordinate vectors) p = (w, 26,31, 2) E R4 which satisfy both these equations simultaneously:
p(0,1,2,3) = O and p(1,2,7,5) = 1. 15 points Turn over 1 2 7. Given: M = < ¥ §> Find all coordinate vectors, 1} = (:3), such that MU = v. 15 points 2 3 8. Let M be an unspeciﬁed matrix, and let v1 and 122 be unspeciﬁed vectors. Let M v1 —
0 and M112 = 0, and let 1);; = 7121 — 5112. Prove that MU3 = 6. 10 points 9. Find all points where the ellipsoid, $2 + 23/2 + 322 2 127 is pierced by the line
x+2y+32=0 and y+22:0. 15pomts How many hours per week do you study for this course? Do you study with a buddy? 10 points
Total 146 points. Homework. For Wednesday: Read Ch. III Sec. 5 and 5B. Browse Sec. 6 Handin Exercises
111.411 and 15, III.5.4(a), III.5B.1(b) Practice. Ch. III Sec. 4 Exercises III.4.12—26, Ch. III Sec. 5 Exercises III.5.1—6, Ch. III Sec.
5B all exercises. For Thursday: Handin Exercise III.4.18 (use results from Exer. III.4.15 and read “set
up”). Exercise III.4.19 (typo ”First ﬁnd x1(t) and y1(t), [later ﬁnd 33(75) and y(t)]), Exercises
III.5.3,4(b),5, Exercise III.5B.2(a) with Example 5B3 as model7 Exercise III.5B.3 and 5. Quiz Friday Looking way ahead: Browse Ch. II Sec. 3 and 4 (on linear equations). Read the material on complex numbers in Appendix C Sec. 5 and 6 and review the material on exponential growth in
Appendix C Sec. 2. ...
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 Spring '08
 Jake

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