Treinamento para
Olimpadas de
Resolues
Matemtica
2008
www.cursoanglo.com.br
NVEL 3
AULAS 7 a 9
Em Classe
1.
1
2
x
1 1 4x 2
1
IR
x
2
x
Note que 1 4 x 2 IR
Logo,
1
.
2
1
e x 0.
2
Note que, nessas condies, 1 + 1 4 x 2
(*)
0.
Multiplicando, ambos os membro

ACM 104
Homework Set 2 Solutions
January 24, 2001
1 Franklin Chapter 1, Problem 7, page 20.
Let A be an n n triangular matrix, with aij = 0 for j < i. Assuming all aii = 0, prove that A1
is also a triangular matrix.
We will do the proof only in the case o

ACM 104
Homework Set 1 Solutions
January 16, 2001
1. Franklin Chapter 1, Problem 2, page
Let A be of the form the form
a11
a21
A= 0
0
0
9.
a12
a22
0
0
0
a13
a23
a33
a43
a53
a14
a24
a34
a44
a54
a15
a25
a35
a45
a55
=
A1
0
.
A2
Show that
det A = det
a11
a2

ACM 104
Homework Set 4 Solutions
February 14, 2001
1 Franklin Chapter 2, Problem 4, page 55.
Suppose that we feel that some observations are more important or reliable than others. Redene
the function to be minimized as:
m
2
i (bi x0 xi ai1 x2 ai2 . . . x

ACM 104
Homework Set 3 Solutions
January 29, 2001
2 Franklin Chapter 2, Problem 2, page 50.
Let A be an mxn matrix, and B be an m p matrix. Give a necessary and sucient condition so
that the system AX = B will have an n p matrix-solution X. If X 0 is a pa

ACM 104
Homework Set 5 Solutions
February 21, 2001
1 Franklin Chapter 4, Problem 4, page 102.
Let A be an n n non-Hermitian matrix. Suppose that A has distinct eigenvalues 1 , . . . , n .
Show that A has the eigenvalues 1 , . . . , n . If uj is an eigenve