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Unformatted text preview: Prof. Dancis MATH 401 Test 1 March 1, 2003 Use the methods of this course. \\'rite in complete sentenses. 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. (Note I = :r(t) and = I” and v = v(t)”.) Use this info when it will shortcut calculations. If A! = 2), then ill—1 2 (:16 jib), whenever det, 1‘! ;£ 0. Proposition on Invertibility. Given an invertible n X nmatrix ill and another 71 X 'nniatrix
E. If/when [111‘1E[ in < 1”, then A! + E is also an invertible matrix. 1. (a) What is the reason or motivation for the deﬁnition of matrix times matrix multiplication.
(A single equation will sufﬁce.) 2 points (b) Given unknown 7 X 7—matrices, A], N and P, where P and It! are invertible matrices,
such that [W = P‘lNP. Prove that N is also invertible, and find a formula for N in terms of
1V and P. 10 points *1 2 1 1
[El < J2. but [A X J;[ < [A X E[. 6 points (c) Let A = (2 1) and J2 = <1 1). State/ﬁnd a speciﬁc 2 X 2niatrix, E, such that 2. (a) Kirchoff’s equations for a particular unspeciﬁed DC (Direct Current) electrical resistance . . . . . . . ~ ~ V Circuit. of two loops containing two batteries w1th voltages V1 and Vg is: RI : V = (V L V 7
v i 2 . . . , , . .  11 where R is an invertible 2 X 2 reSistance matrix and I = < . ’lo ) is the current Hector. (a) Would it be correct to write iR = V instead of Pi = V? \Vhy or why not? 3 points (b) Show that the current vector i is the sum of a current vector i1 due to battery V1 alone
plus a current vector 12 due to battery V2 alone. (5 points 3. (a) Given unknown 7 X 7—matrices. 1U, N. P and D, where P and M are invertible
matrices, such that AI‘IN = P‘IDP. Let v = v(t) E R7 X R be a vector valued function of
time. Let w 2 PV. Given .11v 2 NV. Show that w 2 DW. 10 points —7 O
(i —:3
matrix and S is a symmetric matrix. such that S = PTDP. Let v : (ii 6 R’— be a coordinate
vector. Let = w 2 PV. Given : =f(1'.y) and (3) = VTSV. Show that = WTDW. Show that. f 1.2g) ‘: O,V(I‘y)¢(0_0). 10 points (b) Giver: D: < ) and unknown ‘2 .V 2 lii:‘.iI'lC('S. S and P, where P is an invertible 4. Given that A! is an unspeciﬁed 7 X 7matrix. such that AITAI is invertible. Given
P = AI(A1TAI)‘1AI. Prove that PT = P. Do not quote the theorem that say this is so. State each rule for transposes that you used. 14 points 5. Given two constant 5X5—matrices, K and Ali, and a constant coordinate vector. we 6 R5. Let,
v = v(t) be a vector valued function of time, which satisﬁes the equation: AIR} — Ix'v : sin 7! we.
Find a particular solution, v(t), to this equation which has the form v = v(t) = sin 7: wI, where
WI 6 R5. You may assume that various matrices are invertible when needed. Find v(t) in terms of (the known information) Al, K and wo. 10 points
6, (a) Let, 311; = 11;, “vhere = and ’U = and 111 = Suppose that
Al is exact, and the tolerance for its first coordinate (—3) is 0.2 and the tolerance for its second
coordinate (7) is 0.1. Find good tolerances for the coordinates of v. 15 points
(b) Let A! = (32 and suppose that the tolerance for each entrie in A1 is 1.0. Let E
be the unknown error matrix for 1V1. Prove that M + E is always invertible. . . .. . 8 points How many hours per week do you study for this c0urse? Do you study with a buddy? 4 points
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This note was uploaded on 05/04/2008 for the course MATH 401 taught by Professor Dancis during the Spring '04 term at Maryland.
 Spring '04
 Dancis
 Math, Linear Algebra, Algebra

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