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# Algebra Linear - Luis Fernando - Lista 1

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```ÁLGEBRA LINEAR
LISTA 1
ASSUNTO: ARITMÉTICAS DAS MATRIZES E MATRIZES ESCALONADAS
PROF: LUIZ FERNANDO

1 -Sejam A =
1 1
2 2
1 0
\uf02d\uf0e9
\uf0eb
\uf0ea
\uf0ea
\uf0ea
\uf0f9
\uf0fb
\uf0fa
\uf0fa
\uf0fa
e B =
3 1
1 4\uf02d
\uf0e9
\uf0eb
\uf0ea
\uf0f9
\uf0fb
\uf0fa . Determine uma matriz C tal que CA = B .
2 -Sejam as matrizes A = \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
\uf02d 24
31
, B = \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
\uf02d
\uf02d
031
120
, C =
\uf0fa
\uf0fa
\uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0ea
\uf0ea
\uf0eb
\uf0e9
\uf02d
\uf02d
132
030
102
, D =
\uf0fa
\uf0fa
\uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0ea
\uf0ea
\uf0eb
\uf0e9 \uf02d
33
00
42
, E = \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9 \uf02d
301
211

Verifique quais das seguintes expressões abaixo são definidas, e calcule aquelas que são definidas.

a) 2B-3E, b) AB, c) BA, d) BD, e) DB, f) CD + 3DB.

3 - Sejam A = \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
\uf02d
\uf02d
42
21
, B = \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
31
62
. Calcule AB e BA . São iguais estes produtos ? Justifique.

4- Se A é uma matriz quadrada, então pode ser multiplicada por si mesma, e podemos definir A2 = AA,
A3 =AAA ,..., An = AA...A (n fatores). Encontre A2 e A3 , se a) A = \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
\uf02d 10
12
, b) A =
\uf0fa
\uf0fa
\uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0ea
\uf0ea
\uf0eb
\uf0e9
\uf02d
\uf02d
\uf02d
112
101
101
.
5 \u2013 Mostre que
2
ac
ba
\uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
\uf02d
= \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
10
01
se, somente, se a
2
+ bc = 1

6 \u2013 Quais as matrizes abaixo estão na forma escalonada ? Justifique sua resposta.
i)
3 1 2
2 1 1
1 3 0
\uf02d
\uf02d
\uf0e9
\uf0eb
\uf0ea
\uf0ea
\uf0ea
\uf0f9
\uf0fb
\uf0fa
\uf0fa
\uf0fa
ii) \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
00
00
iii)
\uf0fa
\uf0fa
\uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0ea
\uf0ea
\uf0eb
\uf0e9
\uf02d
800
050
300
iv) \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
\uf02d 041
000

7- Passe para forma escalonada as matrizes acima, que não estão na forma escalonada.

8 \u2013 Sejam M = \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
43
21
, M\u2019 = \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
63
21
, W = \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
5
1
e 0 = \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
0
0
. Determine matrizes V = \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
y
x
e V\u2019 = \uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0eb
\uf0e9
'
'
y
x
tais que
MV = W e MV\u2019 = 0.

9 \u2013 Dada a matriz M =
\uf0fa
\uf0fa
\uf0fa
\uf0fb
\uf0f9
\uf0ea
\uf0ea
\uf0ea
\uf0eb
\uf0e9
\uf02d
5
3
5
4
5
4
5
3
. i) Mostre que M
2
= I2 , ii) Ache números reais a e b, tais que a matriz
P = aM +bI2 cumpra P
2 = P e seja não nula .

PROF: LUIZ FERNANDO```