linear algebra problems, Exercises of Linear Algebra

season 3 to 6 in linear algebra

Typology: Exercises

2022/2023

Uploaded on 05/22/2024

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اﺪمﻧﺎﺑـﻪ
:سدرمﻧﺎ
ﺮﻔﻧﺎﻳﺮﻄﻋ ﺮﺘﻛﺩ :ﺱﺭﺪﻣ
ﻡﻭﺩ ﻱﺮﺳ ﻦﻳﺮﻤﺗ
−−−−−:ﻞﻳﻮﺤﺗ
؟ﺩﻮﺑ ﺪﻫﺍﻮﺧ ﻦﻴﻌﻣ ﺖﺒﺜﻣ ﺮﻳﺯ ﻱﺎﻫ ﺲﻳﺮﺗﺎﻣ kﺯﺍ ﻱﺮﻳﺩﺎﻘﻣ ﻪﭼ ﻱﺍﺯﺍ ﻪﺑ
A=
2 1 k
1 2 1
k1 2
, B =
1k0
k1k
0 1 k
.ﺪﻴﻨﻛ ﻪﺒﺳﺎﺤﻣ ﺍﺭ ﺮﻳﺯ ﻱﺎﻫ ﺲﻳﺮﺗﺎﻣ ﺯﺍ ﻚﻳ ﺮﻫ ﺱﻮﻜﻌﻣ ، ﻲﻗﺎﺤﻟﺍ ﺲﻳﺮﺗﺎﻣ ﻪﺒﺳﺎﺤﻣ ﻖﻳﺮﻃ ﺯﺍ
A=
102
321
510
, B =
tan(x) 0 cot(x)
0 1 0
tan(x) 0 cot(x)
.ﺪﻴﺑﺎﻴﺑ ﺍﺭ ﺮﻳﺯ ﻱﺎﻫ ﺲﻳﺮﺗﺎﻣ ﺖﻳﺎﻬﻨﻴﺑ ﻭﺩ ، ﻚﻳ ﻡﺮﻧ
A=
1 0 2 5
3 2 7 1
5 1 0 9
3 8 2 5
, B =
0 0 0
0 1 0
b0a
؟ﺪﻧﺮﻳﺬﭘ ﺱﻮﻜﻌﻣ ﺮﻳﺯ ﻱﺎﻬﺴﻳﺮﺗﺎﻣ kﺯﺍ ﻱﺮﻳﺩﺎﻘﻣ ﻪﭼ ﻱﺍﺮﺑ
A=
k1 4
0k+ 1 1
0 0 k3
, B =
k1 4
2 0 1
101
:ﻢﻳﺭﺍﺩﻭ ﺪﻧﺮﻳﺬﭘ ﺱﻮﻜﻌﻣ ﻲﮕﻤﻫﻭ ﻪﺑﺎﺸﻣ ﺩﺎﻌﺑﺍ ﺎﺑS،R،Q،Pﻲﻌﺑﺮﻣ ﺲﻳﺮﺗﺎﻣ ﺭﺎﻬﭼ
P=QR1S
.ﺪﻳﺭﻭﺁ ﺖﺳﺩ ﻪﺑ ﻥﺎﺸﺳﻮﻜﻌﻣ ﺎﻳ/ﻭ S،Q،Pﺱﺎﺳﺍﺮﺑ ﺍﺭ Rﺲﻳﺮﺗﺎﻣ
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ﻣﺎﺗﺮﻳﺲ ﻫﺎﻱ ﺯﻳﺮ ﻣﺜﺒﺖ ﻣﻌﻴﻦ ﺧﻮﺍﻫﺪ ﺑﻮﺩ؟ k ﺑﻪ ﺍﺯﺍﻱ چﻪ ﻣﻘﺎﺩﻳﺮﻱ ﺍﺯ -۱

A =

2 1 −k

k − 1 2

, B =

1 k 0

k 1 k

0 1 k

A =

, B =

tan(x) 0 cot(x)

tan(x) 0 cot(x)

A =

, B =

b 0 a

ﻣﺎﺗﺮﻳﺴﻬﺎﻱ ﺯﻳﺮ ﻣﻌﻜﻮﺱ پﺬﻳﺮﻧﺪ؟ kﺑﺮﺍﻱ چﻪ ﻣﻘﺎﺩﻳﺮﻱ ﺍﺯ -۴

A =

k − 1 4

0 k + 1 1

0 0 k − 3

, B =

k − 1 4

ﺑﺎ ﺍﺑﻌﺎﺩ ﻣﺸﺎﺑﻪ ﻭﻫﻤگﻲ ﻣﻌﻜﻮﺱ پﺬﻳﺮﻧﺪ ﻭﺩﺍﺭﻳﻢ:S،R،Q،P چﻬﺎﺭ ﻣﺎﺗﺮﻳﺲ ﻣﺮﺑﻌﻲ -۵

P = QR

− 1 S

ﻭ/ﻳﺎ ﻣﻌﻜﻮﺳﺸﺎﻥ ﺑﻪ ﺩﺳﺖ ﺁﻭﺭﻳﺪ. S،Q،P ﺭﺍ ﺑﺮﺍﺳﺎﺱ Rﻣﺎﺗﺮﻳﺲ

A =

A

⊤ P + P A = −Q

ﻳﻚ ﻣﺎﺗﺮﻳﺲ ﻣﺠﻬﻮﻝ ﺍﺳﺖ. Pn×n ﻣﺘﻘﺎﺭﻥ ﻭ Qn×n ﻣﺎﺗﺮﻳﺲ ﺩﻟﺨﻮﺍﻩ ، An×nﺩﺭﺍﻳﻦ ﻣﻌﺎﺩﻟﻪ

ﻳﻚ ﻣﺎﺗﺮﻳﺲ ﻣﺘﻘﺎﺭﻥ ﺍﺳﺖ. P ﺁ( ﻧﺸﺎﻥ ﺩﻫﻴﺪ

ﺭﺍ ﺑﻴﺎﺑﻴﺪ. P ﺑﻪ ﺷﻜﻞ ﻣﺎﺗﺮﻳﺲ ﻫﺎﻱ ﺯﻳﺮ ﺑﺎﺷﻨﺪ ، Q ﻭ A ﺏ( ﺍگﺮ ﻣﺎﺗﺮﻳﺲ ﻫﺎﻱ

A =

 , Q^ =

ﻏﻴﺮ ﻣﻨﻔﺮﺩ ﺑﺎﺷﻨﺪ ﻧﺸﺎﻥ ﺩﻫﻴﺪ: A + B ﻭ B ، A ﺍگﺮ -۸

A(A + B)

− 1 B = B(A + B)

− 1 A = (A

− 1

  • B

− 1 )

− 1

ﺭﺍ ﺑﺪﻭﻥ ﺣﻞ ﻣﺴﺘﻘﻴﻢ ﺑﻴﺎﺑﻴﺪ. |A|ﺭﺍ ﺑﻪ ﺻﻮﺭﺕ ﺯﻳﺮ ﺗﻔﻜﻴﻚ ﻛﻨﻴﻢ ، A ﺍگﺮ ﺑﺘﻮﺍﻧﻴﻢ -۹

A =

= I 5 +

[

]

(n ≥ 1 ﺍﺳﺖ.) P

− 1 A

n P = B

n

ﺑﺎﺷﺪ ، ﺁﻧگﺎﻩ P

− 1

AP = B ﻧﺸﺎﻥ ﺩﻫﻴﺪ ﺍگﺮ -۱۰

A =

k 1 1 1

k k 1 1

k k k 1

= (1 − k)

3 , B =

1 + k 1 k 1 k 1 k 1

k 2 1 + k 2 k 2 k 2

k 3 k 3 1 + k 3 k 3

k 4 k 4 k 4 1 + k 4

= 1 + k2 + k3 + k 4