Giáo trình Đại số tuyến tính và Ứng dụng

❱■➏◆ ❍⑨◆ ▲❹▼ ❑❍❖❆ ❍➴❈ ❱⑨ ❈➷◆● ◆●❍➏ ❱■➏❚ ◆❆▼
❱■➏◆ ❚❖⑩◆ ❍➴❈

❚→❝ ❣✐↔✿ ◆❣✉②➵♥ ❇➼❝❤ ❱➙♥✱ ✣é ❚❤→✐ ❉÷ì♥❣

✣❸■ ❙➮ ❚❯❨➌◆ ❚➑◆❍ ❱⑨
▼❐❚ ❙➮ ❇⑨■ ❚❖⑩◆ ❚❍Ü❈
❚■➍◆
●✐→♦ tr➻♥❤ ❜➟❝ ✤↕✐ ❤å❝

◆➠♠ 2022


◆●❯❨➍◆ ❇➑❈❍ ❱❹◆✱✣➱ ❚❍⑩■ ❉×❒◆●

✣❸■ ❙➮ ❚❯❨➌◆ ❚➑◆❍ ❱⑨
▼❐❚ ❙➮ ❇⑨■ ❚❖⑩◆ ❚❍Ü❈ ❚■➍◆

◆➠♠ 2022


▲í✐ ♥â✐ ✤➛✉
✣↕✐ sè t✉②➳♥ t➼♥❤ ❧➔ ♠ët ❝❤✉②➯♥ ♥❣➔♥❤ ỡ tr t ồ ợ ố tữủ
ự ❧➔ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤✱ ❝→❝ ♠❛ tr➟♥✱ ❝→❝ ❦❤æ♥❣
❣✐❛♥ ✈❡❝t♦r✱ ❝→❝ →♥❤ ①↕ t✉②➳♥ t➼♥❤✳✳✳ ❚❤✉➟t t♦→♥ ✤➸ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥
t➼♥❤ ♠➔ ♥❣➔② ♥❛② ♥❣÷í✐ t❛ ❣å✐ ❧➔ ♣❤➨♣ ❦❤û ●❛✉ss ✭①❡♠ ♣❤➛♥ ✷✳✷✳ ✈➔
t tứ rt sợ ữủ tr tr ❝❤÷ì♥❣ ✽ ✧P❤÷ì♥❣ tr➟♥✧ ❝õ❛ ❝✉è♥
s→❝❤ ✧❈û✉ ❝❤÷ì♥❣ t♦→♥ t❤✉➟t✧ ❝õ❛ ♥❣÷í✐ ❚r✉♥❣ ◗✉è❝ ✤÷đ❝ ❜✐➯♥ s♦↕♥ ✈➔♦ t❤í✐
✣ỉ♥❣ ❍→♥ ✭❝â t➔✐ ❧✐➺✉ ❝❤♦ r➡♥❣✱ ♥â ✤÷đ❝ ✈✐➳t ✈➔♦ ❦❤♦↔♥❣ trữợ ổ
r s õ ữủ ✈✐➳t ❜ê s✉♥❣ ❜ð✐ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ❚r✉♥❣
◗✉è❝✱ tr♦♥❣ ✤â ❝â ▲÷✉ ❍✉② ✈➔ ❚ê ❳✉♥❣ ❈❤✐✮✳ ❚r♦♥❣ ❝❤÷ì♥❣ ✽ ❝õ❛ ❝✉è♥ s→❝❤

♥➔②✱ ❤å ✤➣ tr➻♥❤ ❜➔② 18 ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ s↔♥ ❧÷đ♥❣ ♥ỉ♥❣ ♥❣❤✐➺♣ ✈➔ ✈✐➺❝
♠✉❛ ❜→♥ ❣✐❛ só❝✱ ❞➝♥ ✤➳♥ ✈✐➺❝ ❣✐↔✐ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤✳ Ð ❝❤➙✉ ❹✉✱
❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ①✉➜t ❤✐➺♥ ❧➛♥ ✤➛✉ t✐➯♥ ✈➔♦ ♥➠♠ ✶✻✸✼ ❦❤✐ ❘➨♥❡
❉❡s❝❛rt❡s ✤÷❛ ✈➔♦ ❦❤→✐ ♥✐➺♠ tå❛ ✤ë tr♦♥❣ ❤➻♥❤ ❤å❝✳ ❚r♦♥❣ ❤➻♥❤ ❤å❝ ❉❡s❝❛rt❡s✱
❝→❝ ✤÷í♥❣ t❤➥♥❣ ✈➔ ❝→❝ ♠➦t ♣❤➥♥❣ ✤÷đ❝ ♠ỉ t↔ ❜➡♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥
t➼♥❤ ✈➔ ✈✐➺❝ t➻♠ ❣✐❛♦ ❝õ❛ ❝❤ó♥❣ ❞➝♥ ✤➳♥ ✈✐➺❝ ❣✐↔✐ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥
t➼♥❤ ✭①❡♠ ❝→❝ ❱➼ ❞ư ✷✳✹✱ ✷✳✼✱✷✳✽✮✳ P❤÷ì♥❣ ♣❤→♣ sû ❞ư♥❣ ✤à♥❤ t❤ù❝ ✤➸ ❣✐↔✐
❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ✤➛✉ t✐➯♥ ❜ð✐ ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠
▲❡✐❜♥✐③ ✈➔♦ ♥➠♠ ✶✻✾✸✳ ❙❛✉ ✤â✱ ✈➔♦ ♥➠♠ ✶✼✺✵ ●❛❜r✐❡❧ ❈r❛♠❡r ✤➣ ✤÷❛ r❛ ❝→❝
❝ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠ ❝❤♦ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t ợ t
ữủ ồ ổ tự rr ỵ õ r rr
ss ✤÷❛ r❛ ♣❤÷ì♥❣ ♣❤→♣ ❦❤û ✤➸ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤✱ ♣❤÷ì♥❣
♣❤→♣ ♥➔② ❦❤✐ ✤â ✤÷đ❝ ❝♦✐ ❧➔ ♠ët ữợ t ợ tr tr
s s ❙②❧✈❡st❡r ✤➣ ✤÷❛ ✈➔♦ ❦❤→✐ ♥✐➺♠ ♠❛ tr➟♥ ✭✧♠❛tr✐①✧ tr♦♥❣ t✐➳♥❣
▲❛t✐♥✱ ①✉➜t ①ù tø t✐➳♥❣ ✧♠❛t❡r✧ ❝â ♥❣❤➽❛ ❧➔ ✧♠➭✧✮✳ ➷♥❣ ①❡♠ ♠❛ tr➟♥ ♥❤÷ ❧➔
✤è✐ t÷đ♥❣ s✐♥❤ r❛ ❝→❝ ✤à♥❤ t❤ù❝ ♠➔ ♥❣➔② ♥❛② t❛ ❣å✐ ❧➔ ♠✐♥♦r ✭①❡♠ ✣à♥❤ ♥❣❤➽❛
✶✳✶✺✮✳ ❙❛✉ ✤â✱ ✈➔♦ ♥➠♠ ✶✽✺✻✱ ❆rt❤✉r ❈❛②❧❡② ✤➣ ✤à♥❤ ♥❣❤➽❛ ❝→❝ ♣❤➨♣ t♦→♥ ❝ë♥❣✱
trø✱ ♥❤➙♥ ♠❛ tr➟♥✱ ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝
♣❤➨♣ t♦→♥ ♥➔② ✭①❡♠ ♣❤➛♥ ✶✳✷✳✱ ✶✳✸✳✱ ✶✳✻✳✮✳ ✣à♥❤ ♥❣❤➽❛ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝t♦r ✤÷đ❝
P❡❛♥♦ ✤÷❛ r❛ ✈➔♦ ♥➠♠ ✶✽✽✽ ✭①❡♠ ữỡ s õ ỵ tt
t✉②➳♥ t➼♥❤ ❣✐ú❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝t♦r ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✤➣ r❛ ✤í✐ ✭①❡♠


❈❤÷ì♥❣ ✺✮✳ ◆❣➔② ♥❛②✱ ✤↕✐ sè t✉②➳♥ t➼♥❤ ♥❣➔② ❝➔♥❣ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣
tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❝õ❛ ❦❤♦❛ ❤å❝ ❦ÿ t❤✉➟t ✈➔ ❝ỉ♥❣ ♥❣❤➺✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷ t❛ ❝â
t❤➸ ù♥❣ ❞ư♥❣ ❝→❝ ♣❤➨♣ t♦→♥ ✈ỵ✐ ♠❛ tr➟♥ tr♦♥❣ tt t ỷ ỵ t
ố ũ tr ✤➸ ♠ỉ t↔ ❝→❝ ♠ỉ ❤➻♥❤ t❤í✐ t✐➳t✱ ♠ỉ ❤➻♥❤ ❞✐ ❝÷✱♠➟t
♠➣✳✳✳✭①❡♠ ♣❤➛♥ ✶✳✼✳✮✱ ❞ị♥❣ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✤➸ ♣❤➙♥ t➼❝❤ ❣✐❛♦
t❤æ♥❣✱ ♣❤➙♥ t➼❝❤ ♠↕❝❤ ✤✐➺♥ ✭①❡♠ ♣❤➛♥ ✷✳✻✳✷✳✮✱ ❞ü ❜→♦ t➠♥❣ tr÷ð♥❣ ❞➙♥ sè✱ ①➙②
❞ü♥❣ ♠ỉ ❤➻♥❤ tr♦♥❣ t❤✐➯♥ ✈➠♥ ❤å❝ ✭①❡♠ ♣❤➛♥ ✹✳✸✳✹✳✮✱ ❞ò♥❣ →♥❤ t t

tr ỗ ồ t ❞ị♥❣ ❣✐→ trà r✐➯♥❣ ✈➔ ✈❡❝t♦r r✐➯♥❣ ✤➸
❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤✱ ①➙② ❞ü♥❣ t❤✉➟t t♦→♥ ❙❱❉ ✤➸ ❣✐↔♠
❝❤✐➲✉ ❞ú ❧✐➺✉ ✭①❡♠ ♣❤➛♥ ✺✳✺✳✮✳✳✳
●✐→♦ tr➻♥❤ ♥➔② ✤÷đ❝ ❝❤ó♥❣ tỉ✐ ❜✐➯♥ s♦↕♥ ❞ü❛ tr➯♥ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦
❬✶❪✱ ❬✷❪ ✈➔ ❝→❝ ❜➔✐ ❣✐↔♥❣ ❝õ❛ ❝❤ó♥❣ tỉ✐ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❈ỉ♥❣ ♥❣❤➺✲✣❍ ◗✉è❝
❣✐❛ ❍➔ ◆ë✐ ✈➔ ✣↕✐ ❤å❝ ❱✐➺t✲P❤→♣✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ❣✐→♦ tr➻♥❤ ❧➔ ❣✐ỵ✐ t❤✐➺✉ ✤➳♥
❝→❝ s✐♥❤ ✈✐➯♥✱ ❝→❝ ❤å❝ ✈✐➯♥ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❝õ❛ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✈➔ ❝→❝
ù♥❣ ❞ö♥❣ t❤ü❝ t✐➵♥ ❝õ❛ ❝❤ó♥❣ tr♦♥❣ ❦❤♦❛ ❤å❝ ✈➔ ✤í✐ sè♥❣✳
◆❤â♠ t→❝ ❣✐↔ tr➙♥ trå♥❣ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❱✐➺♥ ❚♦→♥ ❤å❝✲❱✐➺♥ ❍➔♥ ❧➙♠
❑❤♦❛ ❤å❝ ✈➔ ❈æ♥❣ ♥❣❤➺ ❱✐➺t ◆❛♠✱ ◗✉ÿ ✣ê✐ ♠ỵ✐ s→♥❣ t↕♦ ❱■◆■❋ ✤➣ t↕♦ ♠å✐
✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï ✤➸ ❣✐→♦ tr➻♥❤ ✤÷đ❝ ①✉➜t ❜↔♥✳
▼➦❝ ❞ị ✤➣ r➜t ❝è ❣➢♥❣✱ ♥❤÷♥❣ tr♦♥❣ q✉→ tr➻♥❤ ❜✐➯♥ s♦↕♥✱ ❝❤ó♥❣ tỉ✐ ❦❤ỉ♥❣
t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❈❤ó♥❣ tỉ✐ r➜t ♠♦♥❣ ố ữủ ỳ ỵ
õ õ ở tự ừ ở ỗ ồ
ỵ õ õ ỷ t ❞t❞✉♦♥❣❅♠❛t❤✳❛❝✳✈♥✳
❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✳
❍➔ ◆ë✐✱ ♥❣➔② ✷✵ t❤→♥❣ ✷ ♥➠♠ ✷✵✷✶

◆❤â♠ t→❝ ❣✐↔✿

◆❣✉②➵♥ ❇➼❝❤ ❱➙♥ ✈➔ ✣é ❚❤→✐ ❉÷ì♥❣

✹


◆❤ú♥❣ ❦➼ ❤✐➺✉
❚r♦♥❣ ❣✐→♦ tr➻♥❤ ♥➔②✱ t❛ ❞ò♥❣ ♥❤ú♥❣ ❦➼ ợ ỵ tr
ữợ

N

N
Z
R

AB
AB

t ❤ñ♣ sè tü ♥❤✐➯♥
t➟♣ ❤ñ♣ sè tü ♥❤✐➯♥ ❦❤→❝ 0
t➟♣ ❤ñ♣ sè ♥❣✉②➯♥
t➟♣ ❤ñ♣ sè t❤ü❝
t➟♣ ❤ñ♣ ré♥❣
❣✐❛♦ ❝õ❛ t➟♣ ❤ñ♣ A ✈➔ B
❤ñ♣ ❝õ❛ t➟♣ ❤ñ♣ A ✈➔ B


▼ư❝ ❧ư❝
▲í✐ ♥â✐ ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
◆❤ú♥❣ ❦➼ ❤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼ö❝ ❧ö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈❤÷ì♥❣ ✶✳ ▼❛ tr➟♥ ✈➔ ✤à♥❤ t❤ù❝

✸
✺
✻

✶✵

✶✳✶✳ ●✐ỵ✐ t❤✐➺✉ ✈➲ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✷✳ ❈→❝ ♣❤➨♣ t♦→♥ ✈ỵ✐ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✶✳✷✳✶✳ P❤➨♣ ❝ë♥❣ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✶✳✷✳✷✳ P tr ợ ởt số ợ ổ ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✶✳✷✳✸✳ P❤➨♣ ♥❤➙♥ ❤❛✐ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✶✳✸✳ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ♣❤➨♣ t♦→♥ ✈ỵ✐ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✶✳✹✳ P❤➨♣ ❝❤✉②➸♥ ✈à ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✶✳✺✳ ▼❛ tr➟♥ ❞↕♥❣ ❜➟❝ t❤❛♥❣ t❤❡♦ ❞á♥❣ ✈➔ ♣❤➨♣ ❦❤û ●❛✉ss ✳ ✳ ✳ ✳ ✳ ✷✹
✶✳✻✳ ▼❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ♠ët ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✶✳✻✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✶✳✻✳✷✳ P❤➨♣ ❦❤û ●❛✉ss✲❏♦r❞❛♥ ✤➸ t➻♠ ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ ❝õ❛
♠ët ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✶✳✻✳✸✳ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✶✳✼✳ ▼ët sè ù♥❣ ❞ö♥❣ ❝õ❛ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✶✳✼✳✶✳ ▼❛ tr➟♥ tr♦♥❣ t❤✉➟t t♦→♥ ❧➔♠ ❣✐↔♠ ❝❤➜t ❧÷đ♥❣ ❤➻♥❤ ↔♥❤

✸✷

✶✳✼✳✷✳ ▼❛ tr➟♥ tr♦♥❣ t❤✉➟t t♦→♥ ❧➔♠ ♠í ↔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✶✳✼✳✸✳ ▼❛ tr➟♥ tr♦♥❣ t➻♠ ❦✐➳♠ ✤♦↕♥ ♥❤↕❝ ❣è❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✶✳✼✳✹✳ ❈❤✉é✐ ▼❛r❦♦✈✱ ♠❛ tr➟♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ❝→❝ ♠æ ❤➻♥❤ ♣❤ê
❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✶✳✼✳✺✳ ▼❛ tr➟♥ tr♦♥❣ ♠➟t ♠➣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✶✳✼✳✻✳ tr tr t ỗ q ữỡ ọ ♥❤➜t ✳ ✹✸


✶✳✽✳ ✣à♥❤ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✶✳✽✳✶✳ ✣à♥❤ t❤ù❝ ❝õ❛ ♠ët ♠❛ tr➟♥ ✈✉æ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✶✳✽✳✷✳ ❉ị♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ sì ❝➜♣ t❤❡♦ ❞á♥❣ ❤♦➦❝ t❤❡♦ ❝ët
✤➸ t➼♥❤ ✤à♥❤ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✶✳✽✳✸✳ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ✤à♥❤ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✶✳✽✳✹✳ ▼ët sè ù♥❣ ❞ö♥❣ ❝õ❛ ✤à♥❤ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

❈❤÷ì♥❣ ✷✳ ❍➺ ♣❤÷ì♥❣ tr t t



ợ t ữỡ tr ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✼✷
✷✳✶✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
✷✳✶✳✷✳ ❍➺ ❞↕♥❣ ❜➟❝ t❤❛♥❣ ✈➔ ❝→❝❤ ❣✐↔✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼
✷✳✷✳ ✣÷❛ ✶ ❤➺ ♣❤÷ì♥❣ tr t ý t tữỡ ữỡ
ợ ♥â ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽
✷✳✸✳ P❤➨♣ ❦❤û ●❛✉ss ✈➔ ♣❤➨♣ ❦❤û ●❛✉ss✲❏♦r❞❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶
✷✳✹✳ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼
✷✳✺✳ ❱✐➳t ❧↕✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t t ữợ tr ✳ ✽✽
✷✳✻✳ ❈→❝ ù♥❣ ❞ư♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺
✷✳✻✳✶✳ ❚➻♠ ✤÷í♥❣ ❝♦♥❣ ✤❛ t❤ù❝ ♣❤ị ❤đ♣ ✤✐ q
trữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺
✷✳✻✳✷✳ P❤➙♥ t➼❝❤ ♠↕♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻

❈❤÷ì♥❣ ✸✳ ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝t♦r

✶✵✸

✸✳✶✳ ❑❤ỉ♥❣ ❣✐❛♥ Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✸
✸✳✶✳✶✳ ❱❡❝t♦r tr♦♥❣ ♠➦t ♣❤➥♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✸
✸✳✶✳✷✳ ❱❡❝t♦r tr♦♥❣ Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✺
✸✳✶✳✸✳ ❈→❝ ♣❤➨♣ t♦→♥ tr♦♥❣ Rn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼
✸✳✷✳ ❑❤æ♥❣ ❣✐❛♥ ✈❡❝t♦r

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵

✸✳✸✳ ❑❤æ♥❣ ❣✐❛♥ ❝♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷
✸✳✹✳ ❚➟♣ s✐♥❤ ✈➔ sü ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✽
✸✳✺✳ ❈ì sð ✈➔ sè ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✹
✸✳✻✳ ❚å❛ ✤ë ✈➔ ❝❤✉②➸♥ ❝ì sð ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✵
✸✳✼✳ ❍↕♥❣ ❝õ❛ ♠❛ tr➟♥ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✸
✸✳✼✳✶✳ ❍↕♥❣ ❝õ❛ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✸
✼


✸✳✼✳✷✳ ❑❤æ♥❣ ❣✐❛♥ ❤↕❝❤ ❝õ❛ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✼
✸✳✽✳ Ù♥❣ ❞ư♥❣ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝t♦r tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✳ ✳ ✶✹✶

❈❤÷ì♥❣ ✹✳ ❑❤ỉ♥❣ ❣✐❛♥ t➼❝❤ tr♦♥❣
✹✳✶✳ ❚➼❝❤ ❝❤➜♠ tr♦♥❣ R

n

✶✹✾

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✾

✹✳✷✳ ❑❤æ♥❣ ❣✐❛♥ t➼❝❤ tr♦♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✷
✹✳✷✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✷
✹✳✷✳✷✳ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ t➼❝❤ tr♦♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✹
✹✳✷✳✸✳ ❍➻♥❤ ❝❤✐➳✉ ✈✉æ♥❣ ❣â❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✻
✹✳✸✳ ❈ì sð trü❝ ❝❤✉➞♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ t➼❝❤ tr♦♥❣ ✈➔ q✉→ tr➻♥❤ trü❝
❝❤✉➞♥ ❤â❛ ●r❛♠✲❙❝❤♠✐❞t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✾
✹✳✸✳✶✳ ❈ì sð trü❝ ❝❤✉➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✾
✹✳✸✳✷✳ ◗✉→ tr➻♥❤ trü❝ ❝❤✉➞♥ ❤â❛ ●r❛♠✲❙❝❤♠✐❞t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✸
✹✳✸✳✸✳ ❑❤æ♥❣ ❣✐❛♥ ❝♦♥ trü❝ ❣✐❛♦ ✈➔ ❜➔✐ t♦→♥ ❜➻♥❤ ♣❤÷ì♥❣ ♥❤ä

♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✻
✹✳✸✳✹✳ ❈→❝ ù♥❣ ❞ư♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❜➻♥❤ ♣❤÷ì♥❣ ♥❤ä ♥❤➜t ✳ ✳ ✳ ✳ ✶✼✺

❈❤÷ì♥❣ ✺✳ ⑩♥❤ ①↕ t✉②➳♥ t➼♥❤

✶✼✾

✺✳✶✳ ●✐ỵ✐ t❤✐➺✉ ✈➲ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✾
✺✳✶✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✾
✺✳✶✳✷✳ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✵
✺✳✶✳✸✳ ⑩♥❤ ①↕ t✉②➳♥ t➼♥❤ ❝❤♦ ❜ð✐ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✶
✺✳✷✳ ❷♥❤ ✈➔ ❤↕t ♥❤➙♥ ❝õ❛ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✶
✺✳✷✳✶✳ ❍↕t ♥❤➙♥ ❝õ❛ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✶
✺✳✷✳✷✳ ❷♥❤ ❝õ❛ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✸
✺✳✷✳✸✳ ❍↕♥❣ ✈➔ sè ❦❤✉②➳t ❝õ❛ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✸
✺✳✷✳✹✳ ✣ì♥ ❝➜✉✱ t♦➔♥ ❝➜✉ ✈➔ ✤➥♥❣ ❝➜✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✹
✺✳✸✳ ▼❛ tr➟♥ ❝õ❛ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✻
✺✳✸✳✶✳ ▼❛ tr➟♥ ❝❤✉➞♥ t➢❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✻
✺✳✸✳✷✳ ❍ñ♣ t❤➔♥❤ ❝õ❛ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✼
✺✳✸✳✸✳ ⑩♥❤ ①↕ t✉②➳♥ t➼♥❤ ❦❤↔ ♥❣❤à❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✽
✺✳✸✳✹✳ ▼❛ tr➟♥ ❝õ❛ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ✤è✐ ✈ỵ✐ ❝➦♣ ❝ì sð ✳ ✳ ✳ ✳ ✳ ✶✽✾
✽


tr ỡ s tr ỗ ❞↕♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✵
✺✳✺✳ Ù♥❣ ❞ö♥❣ ❝õ❛ →♥❤ ①↕ t✉②➳♥ t➼♥❤ tr ỗ ồ t ✶✾✷

❈❤÷ì♥❣ ✻✳ ❈❤➨♦ ❤â❛ ♠❛ tr➟♥

✷✵✵

✻✳✶✳ ●✐→ trà r✐➯♥❣ ✈➔ ✈❡❝t♦r r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✵
✻✳✶✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✵
✻✳✶✳✷✳ ❈→❝❤ t➻♠ ❣✐→ trà r✐➯♥❣✱ ❝→❝ ✈❡❝t♦r r✐➯♥❣ ✈➔ ❦❤æ♥❣ ❣✐❛♥
r✐➯♥❣ t÷ì♥❣ ù♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✶
✻✳✶✳✸✳ ❈→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ ✶ ♠❛ tr➟♥ t❛♠ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✷
✻✳✶✳✹✳ ●✐→ trà r✐➯♥❣ ✈➔ ✈❡❝t♦r r✐➯♥❣ ❝õ❛ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✷✵✸
✻✳✷✳ ❈❤➨♦ ❤â❛ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✺
✻✳✸✳ ❈❤➨♦ ❤â❛ trü❝ ❣✐❛♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✵
✻✳✹✳ Ù♥❣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✹
✻✳✹✳✶✳ ❉ü ❜→♦ t➠♥❣ tr÷ð♥❣ ❞➙♥ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✹
✻✳✹✳✷✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✼
✻✳✹✳✸✳ ❚❤✉➟t t♦→♥ ❙❱❉ ✈➔ ù♥❣ ❞ö♥❣ tr♦♥❣ ❣✐↔♠ ❝❤✐➲✉ ❞ú ❧✐➺✉ ✳ ✷✷✵

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸✸

✾


ữỡ
tr tự
ợ t tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✷✳ ❈→❝ ♣❤➨♣ t♦→♥ ✈ỵ✐ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✶✳✸✳
✶✳✹✳
✶✳✺✳
✶✳✻✳

✶✳✷✳✶✳ P❤➨♣ ❝ë♥❣ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✶✳✷✳✷✳ P❤➨♣ ♥❤➙♥ ♠❛ tr➟♥ ✈ỵ✐ ♠ët số ợ ổ ữợ

P ♥❤➙♥ ❤❛✐ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ♣❤➨♣ t♦→♥ ✈ỵ✐ ♠❛ tr➟♥ ✳ ✳ ✳
P❤➨♣ ❝❤✉②➸♥ ✈à ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼❛ tr➟♥ ❞↕♥❣ ❜➟❝ t❤❛♥❣ t❤❡♦ ❞á♥❣ ✈➔ ♣❤➨♣ ❦❤û
●❛✉ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
▼❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ♠ët ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼
✷✷
✷✹
✷✻

✶✳✻✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✶✳✻✳✷✳ P❤➨♣ ❦❤û ●❛✉ss✲❏♦r❞❛♥ ✤➸ t➻♠ ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦
❝õ❛ ♠ët ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✶✳✻✳✸✳ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✶✳✼✳ ▼ët sè ù♥❣ ❞ö♥❣ ❝õ❛ ♠❛ tr➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✶✳✼✳✶✳ ▼❛ tr➟♥ tr♦♥❣ t❤✉➟t t♦→♥ ❧➔♠ ❣✐↔♠ ❝❤➜t ❧÷đ♥❣ ❤➻♥❤
↔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✼✳✷✳ ▼❛ tr➟♥ tr♦♥❣ t❤✉➟t t♦→♥ ❧➔♠ ♠í ↔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✼✳✸✳ ▼❛ tr➟♥ tr♦♥❣ t➻♠ ❦✐➳♠ ✤♦↕♥ ♥❤↕❝ ❣è❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✼✳✹✳ ❈❤✉é✐ ▼❛r❦♦✈✱ ♠❛ tr➟♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ❝→❝ ♠æ ❤➻♥❤
♣❤ê ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✼✳✺✳ ▼❛ tr➟♥ tr♦♥❣ ♠➟t ♠➣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
tr tr t ỗ q ❜➻♥❤ ♣❤÷ì♥❣ ♥❤ä
♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✷

✸✺
✸✼
✹✵
✹✶
✹✸

✶✳✽✳ ✣à♥❤ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✶✳✽✳✶✳ ✣à♥❤ t❤ù❝ ❝õ❛ ♠ët ♠❛ tr➟♥ ✈✉æ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✽✳✷✳ ❉ị♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ sì ❝➜♣ t❤❡♦ ❞á♥❣ ❤♦➦❝ t❤❡♦
❝ët ✤➸ t➼♥❤ ✤à♥❤ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✽✳✸✳ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ✤à♥❤ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✽✳✹✳ ▼ët sè ù♥❣ ❞ö♥❣ ❝õ❛ ✤à♥❤ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✹

✹✽
✺✷
✺✼


✶✳✶✳ ●✐ỵ✐ t❤✐➺✉ ✈➲ ♠❛ tr➟♥
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❈❤♦ m, n ❧➔ ❤❛✐ sè tü ♥❤✐➯♥✳ ▼ët ♠❛ tr➟♥ ❦➼❝❤ ❝ï m × n ❧➔
♠ët ♠↔♥❣ ✭❤❛② ❜↔♥❣✮ ❤➻♥❤ ❝❤ú t ỗ m ỏ n ởt ữ s


a11 a12 ... a1n
 a21 a22 ... a2n 

,
 ...

... ... ... 
am1 am2 ... amn
tr♦♥❣ ✤â aij ❧➔ ❝→❝ sè t❤ü❝ ✈➔ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝

❤➺ sè ❝õ❛ ♠❛ tr➟♥✳

❍➺ sè aij ♥➡♠ ð ❞á♥❣ t❤ù i ✈➔ ❝ët t❤ù j ✱ ❝→❝ ❝❤➾ sè i, j ❧➛♥ ❧÷đt ✤÷đ❝ ❣å✐ ❧➔
❝õ❛ aij

❝❤➾ sè ❞á♥❣✱ ❝❤➾ sè ❝ët

♠❛

▼ët ♠❛ tr➟♥ ❝â sè ❞á♥❣ ❜➡♥❣ sè ❝ët ✭tù❝ ❧➔ m = n✮ ✤÷đ❝ ❣å✐ ❧➔ ♠ët
n✳ ❚r♦♥❣ ♠ët ♠❛ tr➟♥ ✈✉ỉ♥❣✱ ❝→❝ ❤➺ sè aii , i = 1, 2, ..., n✱
✤÷đ❝ ❣å✐ ❧➔ ❝→❝
❝õ❛ ♥â✳

tr➟♥ ✈✉ỉ♥❣ ❝➜♣
❤➺ sè tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤
❱➼ ❞ö ✶✳✶✳ ❛✮ √−23 eπ ❧➔ ♠ët ♠❛ tr➟♥ ✈✉ỉ♥❣ ❝➜♣ 2✳ ❈→❝ ❤➺ sè tr➯♥ ✤÷í♥❣
2

❝❤➨♦ ❝❤➼♥❤ ❝õ❛ ♥â ❧➔
❜✮

1
3
2

0
−7
ln2 sin(π/5)

−2, e2 ✳

❧➔ ♠ët ♠❛ tr➟♥ ❦➼❝❤ ❝ï

2 ì 3

ú ỵ ự tt số tr♦♥❣ ♠→② t➼♥❤ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜➡♥❣ ❝→❝
♠❛ tr➟♥ ✤✐➸♠ ↔♥❤ ❝õ❛ ❝❤ó♥❣✳ ◆â✐ r✐➯♥❣✱ ♠é✐ ❜ù❝ ↔♥❤ ❦➽ t❤✉➟t sè ✤❡♥ tr➢♥❣
✭t➯♥ t✐➳♥❣ ❆♥❤✿ ❛ ❞✐❣✐t❛❧ ❣r❛② s❝❛❧❡ ✐♠❛❣❡✮ ❝â ✤ë ♣❤➙♥ ❣✐↔✐
✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜➡♥❣ ♠ët ♠❛ tr➟♥ ù
số tứ

0



255

mìn



ợ số ởt tr♦♥❣ ❝→❝

●✐→ trà ❝õ❛ ♠é✐ ❤➺ sè t❤➸ ❤✐➺♥ ♠ù❝ ✤ë ✤❡♥ tr➢♥❣ ❝õ❛

✤✐➸♠ ↔♥❤ t↕✐ ✈à tr➼ ✤â✱ ❝❤➥♥❣ ❤↕♥ ❤➺ sè ❜➡♥❣
❤♦➔♥ t♦➔♥✱ ❤➺ sè ❜➡♥❣

m×n

255

0

t❤➸ ❤✐➺♥ ✤✐➸♠ ↔♥❤ ♠➔✉ ✤❡♥

t❤➸ ❤✐➺♥ ✤✐➸♠ ↔♥❤ ♠➔✉ tr➢♥❣ ❤♦➔♥ t♦➔♥✳

❱➼ ❞ư ✶✳✷✳ ❚❛ ❝ị♥❣ ①❡♠ ❜ù❝ ↔♥❤ ✤❡♥ tr➢♥❣ ❝â ✤ë ♣❤➙♥ ❣✐↔✐ 8 × 4 ✤ì♥ ❣✐↔♥
s❛✉✿

✶✶


▼❛ tr➟♥ t÷ì♥❣ ù♥❣ ❧➔



0
 32

 64

 96


128

160

192
224

0
32
64
96
128
160
192
224

0
32
64
96
128
160
192
224


0
32 

64 


96 

128

160

192
224

❱➼ ❞ö ✶✳✸ ✭▼❛ tr➟♥ ❜✐➸✉ ❞✐➵♥ ❝õ❛ ♠ët ✤♦↕♥ ➙♠ t❤❛♥❤✮✳ ❱➲ ♠➦t t♦→♥ ❤å❝✱
♠ët ✤♦↕♥ ➙♠ t❤❛♥❤ ❧➔ ởt ữợ sõ ợ tử t tớ

trà ❝õ❛ ❤➔♠ ❧➔ ❜✐➯♥ ✤ë✳ ❇✐➯♥ ✤ë ❞❛♦ ✤ë♥❣ ❣✐ú❛ ❞÷ì♥❣ ✈➔ ➙♠✳ ▼ù❝ ✤ë
❞÷ì♥❣✱ ➙♠ ♣❤ư t❤✉ë❝ ✈➔♦ ➙♠ ❧÷đ♥❣ ❝õ❛ ➙♠ t❤❛♥❤✳
❚r➯♥ ♠→② t➼♥❤ ❦ÿ t❤✉➟t sè✱ ✤♦↕♥ ➙♠ t❤❛♥❤ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜➡♥❣ ♠ët ❞➣② sè✱
❝→❝ ❣✐→ trà ❝õ❛ ❤➔♠ ❧✐➯♥ tư❝ ✤÷đ❝ ❧➜② ♠➝✉ tr♦♥❣ ❝→❝ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ✤➲✉ ✤➦♥
✭✈➼ ❞ư✿ ✹✹✳✶✵✵ ❧➛♥ ♠ët ❣✐➙②✮✳ ❍❛② ♥â✐ ❝→❝❤ ❦❤→❝✱ ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ t❤➔♥❤ ♠ët
♠❛ tr➟♥ ❞á♥❣ ✭❤♦➦❝ ✈❡❝t♦r ❞á♥❣✮✳

❈→❝ ♠❛ tr➟♥ ❝á♥ ✤÷đ❝ ❞ò♥❣ ✤➸ ♠✐♥❤ ❤å❛ ❝→❝ ❜↔♥❣ t❤è♥❣ ❦➯ sè ❧✐➺✉✱ ữ
tr ử ữợ

ử tr✉♥❣ ❜➻♥❤ ❝→❝ ♠æ♥ ❚♦→♥✱ ❱➠♥✱❆♥❤ ❝õ❛ ♠ët ♥❤â♠
❤å❝ s✐♥❤ ♥❤÷ s❛✉✿
❙❚❚

❍å ✈➔ t➯♥

❚♦→♥

❱➠♥

❆♥❤

✶

◆❣✉②➵♥ ❱➠♥ ❆

✽✳✺

✼✳✸

✾✳✵

✷

❚r➛♥ ❚❤à ❇

✾✳✻

✽✳✵

✼✳✻

✸

✣é ❚❤à ❈

✻✳✼

✽✳✺

✾✳✸

✹

▲➯ ❱➠♥ ❉

✶✵✳✵

✾✳✶

✽✳✷

✺

▼❛✐ ❱➠♥ P

✽✳✽

✽✳✸

✾✳✺

◆➳✉ t❛ ✤→♥❤ sè ❝→❝ ❤å❝ s✐♥❤ tø
tø

1

✤➳♥

3✱

1

✤➳♥

5

✈➔ ✤→♥❤ sè ❝→❝ ♠æ♥ ❚♦→♥✱ ❱➠♥✱❆♥❤

t❤➻ t❛ ❝â t❤➸ ♠✐♥❤ ❤å❛ ❜↔♥❣ sè ❧✐➺✉ tr➯♥ ❜➡♥❣ ♠❛ tr➟♥ ❦➼❝❤ ❝ï

✶✷

5×3


s❛✉ ✤➙②



8.5
 9.6

A=
 6.7
10.0
8.8

tr♦♥❣ ✤â ❤➺ sè

aij

7.3
8.0
8.5
9.1
8.3


9.0
7.6

9.3
,
8.2
9.5

t❤➸ ❤✐➺♥ ✤✐➸♠ tr✉♥❣ ❜➻♥❤ ♠æ♥ t❤ù

j

❝õ❛ ❤å❝ s✐♥❤ t❤ù

i

tr♦♥❣

❞❛♥❤ s→❝❤✳

✶✳✷✳ ❈→❝ ♣❤➨♣ t♦→♥ ✈ỵ✐ ♠❛ tr➟♥
✣➸ ❦➼ ❤✐➺✉ ♠❛ tr➟♥✱ t❛ ❞ị♥❣ ❝→❝ ❝→❝❤ s❛✉
✶✳ ❉ò♥❣ ❝→❝ ❝❤ú ❝→✐ ▲❛t✐♥ ✐♥ ❤♦❛✿ A, B, C...
✷✳ ❱✐➳t ♣❤➛♥ tû ✤↕✐ ❞✐➺♥ tr♦♥❣ ♥❣♦➦❝ ✈✉æ♥❣ ❤♦➦❝ ♥❣♦➦❝ trá♥✿ [aij ], [bij ], [cij ]...
✭❤♦➦❝ (aij ), (bij ), (cij )...✮✳


a11 a12 ... a1n
 a21 a22 ... a2n 
 , ❤♦➦❝
✸✳ ❇➡♥❣ ♠ët ♠↔♥❣ ❤➻♥❤ ❝❤ú ♥❤➟t 
 ...
... ... ... 
am1 am2 ... amn


a11 a12 ... a1n
 a21 a22 ... a2n 


 ...
... ... ... 
am1 am2 ... amn

✣à♥❤ ♥❣❤➽❛ ✶✳✷ ✭❍❛✐ ♠❛ tr➟♥ ❜➡♥❣ ♥❤❛✉✮✳ ❍❛✐ ♠❛ tr➟♥ A = [a

✈➔ B = [bij ]
✤÷đ❝ ❣å✐ ❧➔ ❤❛✐ ♠❛ tr➟♥ ❜➡♥❣ ♥❤❛✉ ♥➳✉ ❝❤ó♥❣ ❝â ❝ị♥❣ ❦➼❝❤ ❝ï m × n ✈➔
aij = bij ∀i = 1, m, ∀j = 1, n.

ij ]

❑❤✐ ✤â t❛ ✈✐➳t✿ A = B ✳

❱➼ ❞ö ✶✳✺✳ ❳➨t ❝→❝ ♠❛ tr➟♥ s❛✉✿
A=

1 2
1
,B =
3 4
3

C = 1 3 ,D =
❚❛ t❤➜② ❝→❝ ♠❛ tr➟♥

B

✈➔

C

A

✈➔

B

1 2
x 4

❦❤æ♥❣ ❜➡♥❣ ♥❤❛✉ ✈➻ ❝❤ó♥❣ ❦❤ỉ♥❣ ❝â ❝ị♥❣ ❦➼❝❤ ❝ï✱

❝ơ♥❣ ❦❤ỉ♥❣ ❜➡♥❣ ♥❤❛✉ ✈➻ ❝❤ó♥❣ ❦❤ỉ♥❣ ❝ị♥❣ ❦➼❝❤ ❝ï✱

❜➡♥❣ ♥❤❛✉ ✈➻ ❝❤ó♥❣ ❦❤ỉ♥❣ ❝â ❝ị♥❣ ❦➼❝❤ ❝ï✳

✶✸

B

✈➔

D

❦❤ỉ♥❣


A=D

❦❤✐ ✈➔ ❝❤➾ ❦❤✐

x = 3✳

✶✳✷✳✶✳ P❤➨♣ ❝ë♥❣ ♠❛ tr➟♥
✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ◆➳✉ A = [a

✈➔ B = [bij ] ❧➔ 2 ♠❛ tr➟♥ ❝â ❝ị♥❣ ❦➼❝❤ ❝ï
m × n✱ t❤➻ tê♥❣ ❝õ❛ ❝❤ó♥❣✱ ❦➼ ❤✐➺✉ ❧➔ A + B ởt tr ù m ì n
ữủ ✤à♥❤ ♥❤÷ s❛✉✿

A + B = [aij + bij ]
ij ]

◆➳✉ A ✈➔ B ❧➔ 2 ♠❛ tr➟♥ ❦❤ỉ♥❣ ❝ị♥❣ ❦➼❝❤ ❝ï✱ t❤➻ tê♥❣ ❝õ❛ ❝❤ó♥❣ ❦❤ỉ♥❣ ①→❝
✤à♥❤✳

❱➼ ❞ư ✶✳✻✳

❛✮

❜✮ ❚ê♥❣ ❝õ❛

0 1 −2
2 −3 4
2 −2 2
+
=
3 −5 6
−1 6 −4
2 1 2
 2

0
1
3
−2 1
6  ❦❤æ♥❣ ①→❝ ✤à♥❤ ✈➻ ❝❤ó♥❣
✈➔ −5
3 7
8 −9

❦❤ỉ♥❣ ❝ị♥❣

❦➼❝❤ ❝ï✳

✶✳✷✳✷✳ P❤➨♣ ♥❤➙♥ ♠❛ tr ợ ởt số ợ ổ ữợ
số tỹ ỏ ữủ ồ ổ ữợ
A = [a ] ❧➔ ♠ët ♠❛ tr➟♥ ❦➼❝❤ ❝ï m × n✱ c ❧➔ ♠ët sè
ij

t❤ü❝✳ ❚➼❝❤ ❝õ❛ c ✈➔ A✳ ❦➼ ❤✐➺✉ ❧➔ cA ❧➔ ♠ët ♠❛ tr➟♥ ❦➼❝❤ ù m ì n ữủ
cA = [caij ] ✭tù❝ ❧➔ ♠é✐ ❤➺ sè ❝õ❛ A ✤➲✉ ✤÷đ❝ ♥❤➙♥ ✈ỵ✐ c✮✳

❚❛ ❞ị♥❣ ❦➼ ❤✐➺✉ −B t❤❛② ❝❤♦ (−1)B ✳ ◆➳✉ A = [aij ] ✈➔ B = [bij ] ❧➔ 2 ♠❛
tr➟♥ ❝â ❝ò♥❣ ❦➼❝❤ ❝ï✱ t❛ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ ❤✐➺✉ ❝õ❛ A ✈➔ B ♥❤÷ s❛✉✿

❱➼ ❞ö ✶✳✼✳

A − B := A + (−B) = A + (−1)B = [aij − bij ]




1 2 4
2
0 0
−4 3 .
❈❤♦ A = −3 0 −1 , B =  1
2 1 2
−1 3 2

−B ✈➔ 3A − B ✳

 

3×1
3×2
3×4
3 6 12
●✐↔✐✿ ❛✮ 3A = 3 × (−3) 3 × 0 3 × (−1) = −9 0 −3
3×2
3×1
3×2
6 3 6

 

2
0 0
−2 0
0



1 −4 3 = −1 4 −3
❜✮ −B = (−1)
−1 3 2
1 −3 −2
❚➼♥❤ ❛✮

3A✱

❜✮

✶✹



❝✮

3A − B = 3A + (−B) =



 
3 − 2 6 + 0 12 + 0
1 6
−9 − 1 0 + 4 −3 − 3 = −10 4
6+1 3−3 6−2
7 0




3 6 12
−2 0
0
−9 0 −3 + −1 4 −3 =
6 3 6
1 −3 −2

12
−6
4

✶✳✷✳✸✳ P❤➨♣ ♥❤➙♥ ❤❛✐ ♠❛ tr➟♥
✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ❈❤♦ A = [a ] ❧➔ ♠ët ♠❛ tr➟♥ ❦➼❝❤ ❝ï m × n✱ B = [b

❧➔
♠ët ♠❛ tr➟♥ ❦➼❝❤ ❝ï n × p✳ ❚➼❝❤ ❝õ❛ A ✈➔ B ✱ ❦➼ ❤✐➺✉ ❧➔ AB ✱ ❧➔ ♠ët ♠❛ tr➟♥
❦➼❝❤ ❝ï m × p✿ AB = [cij ] ✈ỵ✐
ij

ij ]

n

✭✶✳✶✮

aik bkj .

cij = ai1 b1j + ai2 b2j + ... + ain bnj =
k=1

❚ù❝ ❧➔✱ ✤➸ t➼♥❤ ❤➺ sè ♥➡♠ ð ❣✐❛♦ ❝õ❛ ❞á♥❣ t❤ù i ✈➔ ❝ët t❤ù j ❝õ❛ t➼❝❤ AB ✱ t❛
❧➜② ❝→❝ ❤➺ sè ð ❞á♥❣ t❤ù i ❝õ❛ ♠❛ tr➟♥ A ♥❤➙♥ ✈ỵ✐ ❝→❝ ❤➺ sè t÷ì♥❣ ù♥❣ ð ❝ët
t❤ù j ❝õ❛ ♠❛ tr B rỗ ở t ợ

ú þ ✶✳✷✳ ◆➳✉ sè ❝ët ❝õ❛ ♠❛ tr➟♥ A ❦❤æ♥❣ ❜➡♥❣ sè ❞á♥❣ ❝õ❛ ♠❛ tr➟♥ B✱
t❤➻ t➼❝❤

AB

❦❤æ♥❣ ①→❝ ✤à♥❤✳

❱➼ ❞ö ✶✳✽✳ ❈❤♦
●✐↔✐✿ AB
❝â ❦➼❝❤ ❝ï



−1 3
−3 2
A =  4 −2 , B =
.
−4 1
5
0

①→❝ ✤à♥❤ ✈➻ sè ❝ët ❝õ❛

3 × 2✱ B

❝â ❦➼❝❤ ❝ï

2 × 2✱

A

❚➼♥❤

AB ✳

B ✭✤➲✉ ❜➡♥❣ 2✮✳ ❱➻ A
♥❣❤➽❛ AB ❝â ❦➼❝❤ ❝ï 3 × 2✿

❜➡♥❣ sè ❞á♥❣ ❝õ❛

♥➯♥ t❤❡♦ ✤à♥❤



c11 c12
AB = c21 c22 
c31 c32
❚r♦♥❣ ✈➼ ❞ö ♥➔②✱ t❛ s➩ ✈✐➳t ❝❤✐ t✐➳t ❝→❝❤ t➼♥❤ tø♥❣ ❤➺ sè ❝õ❛
t❤ù❝

✭✶✳✶✮

AB

t❤❡♦ ❝æ♥❣

✤➸ ❝→❝ s✐♥❤ ✈✐➯♥ ❤✐➸✉ rã✳ ❙❛✉ ♥➔②✱ tr♦♥❣ t❤ü❝ ❤➔♥❤✱ ❝→❝ s✐♥❤ ✈✐➯♥ ❝â

t❤➸ ✈✐➳t ❧✉æ♥ ❦➳t q✉↔ t➼❝❤ ❝→❝ ♠❛ tr➟♥✱ ❦❤æ♥❣ ❝➛♥ ✈✐➳t rã ✈✐➺❝ t➼♥❤ ❝❤✐ t✐➳t tø♥❣
❤➺ sè✳ ❚r♦♥❣ ✈➼ ❞ö ♥➔②✱ t❛ ❝â

n = 2✳

❉♦ ✤â

c11 = a11 b11 + a12 b21 = (−1) × (−3) + 3 × (−4) = −9
c12 = a11 b12 + a12 b22 = (−1) × 2 + 3 × 1 = 1
✶✺


c21 = a21 b11 + a22 b21 = 4 × (−3) + (−2) × (−4) = −4
c22 = a21 b12 + a22 b22 = 4 × 2 + (−2) × 1 = 6
c31 = a31 b11 + a32 b21 = 5 × (−3) + 0 × (−4) = −15
c32 = a31 b12 + a32 b22 = 5 × 2 + 0 × 1 = 10

−9 1
AB =  −4 6  .
−15 10


❱➟②

❈→❝❤ ❣❤✐ ♥❤ỵ ❝ỉ♥❣ t❤ù❝ ✭✶✳✶✮
✣➸ t➼♥❤ ❤➺ sè ð ❣✐❛♦ ❝õ❛ ❞á♥❣ t❤ù i ✈➔ ❝ët t❤ù j ❝õ❛ AB t❛ ❧➜② tø♥❣ ❤➺ sè
❝õ❛ ❞á♥❣ t❤ù i ừ A ợ số tữỡ ự ừ ởt tự j ừ B rỗ ở
t ✈ỵ✐ ♥❤❛✉✳ ✭❙❛✉ ♥➔②✱ ❦❤✐ ❤å❝ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ t➼❝❤ tr t s t
r õ t ổ ữợt ❝❤➜♠✮ ❝õ❛ ❞á♥❣ t❤ù i ❝õ❛ A ✈➔ ❝ët t❤ù
j ❝õ❛ B ✮

cij = (ai1 , ai2 , ..., ain ), (b1j , b2j , ..., bnj ) = ai1 b1j + ai2 b2j + ... + ain bnj

❱➼ ❞ö ✶✳✾✳
❜✮

3 4
−2 5

❛✮



−2 4 2
1 0
3 
−5 7 −1
1 0 0 =
2 −1 −2
−3 6 6
−1 1 −1

1 0
3 4
=
0 1
−2 5
✶✻




❝✮



2
1 −2 −3 −1 = 1
1


❞✮




2
2 −4 −6
−1 1 −2 −3 = −1 2
3
1
1 −2 −3

❚❛ t❤➜② ❝→❝ t➼❝❤ tr♦♥❣ ♠ö❝ ❝✮ ✈➔ ❞✮ ❝õ❛ ✈➼ ❞ö tr➯♥ ❦❤→❝ ♥❤❛✉✳ ✣✐➲✉ ✤â
❝❤ù♥❣ tä ♣❤➨♣ ♥❤➙♥ ♠❛ tr➟♥ ❦❤æ♥❣ ❝â t➼♥❤ ❝❤➜t ❣✐❛♦ ❤♦→♥✳ ❚❛ s➩ trð ❧↕✐ t➼♥❤
❦❤æ♥❣ ❣✐❛♦ ❤♦→♥ ❝õ❛ ♣❤➨♣ ♥❤➙♥ ♠❛ tr➟♥ ð ♣❤➛♥ ✶✳✷✳

✶✳✸✳ ❈→❝ t➼♥❤ t ừ t ợ tr
ỵ ✭❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➨♣ ❝ë♥❣ ♠❛ tr➟♥ ✈➔ ♣❤➨♣ tr ợ
ổ ữợ A, B, C ❝→❝ ♠❛ tr➟♥ ❦➼❝❤ ❝ï m × n✱ c, d ❧➔ ❝→❝ sè t❤ü❝✱ t❤➻
t❛ ❝â
✶✳

A+B =B+A

✷✳

A + (B + C) = (A + B) + C

✸✳

(cd)A = c(dA)

✹✳

1A = A

✺✳

(c + d)A = cA + dA

✻✳

c(A + B) = cA + cB

✭t➼♥❤ ❝❤➜t ❣✐❛♦ ❤♦→♥ ❝õ❛ ♣❤➨♣ ❝ë♥❣ ♠❛ tr➟♥✮
✭t➼♥❤ ❝❤➜t ❦➳t ❤ñ♣ ❝õ❛ ♣❤➨♣ ❝ë♥❣ ♠❛ tr➟♥✮

✭t➼♥❤ ❝❤➜t t ủ ừ tr ợ ổ ữợ

t➼♥❤ ❝❤➜t ✺✮✱✻✮ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ t➼♥❤ ❝❤➜t ♣❤➙♥ ố ừ tr
ợ ổ ữợ

ự ❝→❝ t➼♥❤ ❝❤➜t ♥➔② ❞ü❛ ✈➔♦ ✤à♥❤ ♥❣❤➽❛ ❝→❝ ♣❤➨♣
t♦→♥ ✤è✐ ✈ỵ✐ ♠❛ tr➟♥ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ♣❤➨♣ t♦→♥ ✤è✐ ✈ỵ✐ sè t❤ü❝✳ ❚❛ s➩
❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❝❤➜t ✶✮ ✈➔ t➼♥❤ ❝❤➜t ✻✮✳

❈❤ù♥❣ ♠✐♥❤✳

❈❤♦ A = [aij ], B = [bij ]✳ ❑❤✐ ✤â A + B = [aij + bij ] = [bij + aij ] = B + A.

c(A + B) = [c(aij + bij )] = [caij + cbij ] = cA + cB
❱✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t ❝á♥ ❧↕✐ ①❡♠ ♥❤÷ ❜➔✐ t➟♣ ✈➲ ♥❤➔✳
✶✼


ú ỵ ở tr õ t ❝❤➜t ❦➳t ❤ñ♣✱ ♥➯♥ t❛ ❝â t❤➸ ✈✐➳t
tê♥❣ ❝õ❛ ♥❤✐➲✉ ♠❛ tr➟♥ ♠➔ ❦❤ỉ♥❣ ❝➛♥ sû ❞ư♥❣ ❞➜✉
❚❛ ✈✐➳t

()

✤➸ ♥❤â♠ ❝→❝ ♠❛ tr➟♥✳

A1 + A2 + ... + Ak .

✣à♥❤ ♥❣❤➽❛ ✶✳✻ ✭▼❛ tr➟♥ ❦❤æ♥❣✮✳ ▼❛ tr➟♥ ❦❤æ♥❣ ❦➼❝❤ ❝ï m × n✱ ❦➼ ❤✐➺✉ ❧➔
0mn ✱ ❧➔ ♠ët ♠❛ tr➟♥ ❦➼❝❤ ❝ï m × n ❝â t➜t ❝↔ ❝→❝ ❤➺ sè ✤➲✉ ❜➡♥❣ 0✳

❱➼ ❞ö ✶✳✶✵✳ 0

23

=

0 0 0
0 0 0

❑❤✐ ❦➼❝❤ ❝ï ❝õ❛ ♠❛ tr➟♥ ❦❤æ♥❣ ✤➣ ❜✐➳t✱ t❤➻ t❛ ❝â t❤➸ ❦➼ ❤✐➺✉ ♥â ♠ët ❝→❝❤
✤ì♥ ❣✐↔♥ 0

ỵ t t ừ tr ❦❤ỉ♥❣✮✳ ❈❤♦ A ❧➔ ♠ët ♠❛ tr➟♥ ❦➼❝❤
❝ï

m × n✳

❑❤✐ ✤â✿

✶✳

A + 0mn = 0mn + A = A

✷✳

A + (−A) = 0mn

✸✳ ◆➳✉

cA = 0mn ,

❈❤ù♥❣ ♠✐♥❤✳

t❤➻

c=0

❤♦➦❝

A = 0mn .

❚❛ ❝❤➾ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❝❤➜t ✸✮✳ ●✐↔ sû

cA = 0mn

✭✶✳✷✮

◆➳✉ c = 0✱ t❤➻ t➼♥❤ ❝❤➜t ✸✮ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû c = 0✳ ❚❛ s➩ ❝❤➾ r❛
A = 0mn ✳ ◆❤➙♥ ❝↔ ✷ ✈➳ ❝õ❛ ✤➥♥❣ t❤ù❝ ✭✶✳✷✮ ✈ỵ✐ 1c ✳ ▲ó❝ ✤â ✈➳ tr→✐ ❧➔✿

1
t➼♥❤ t ừ ỵ 1
t t ừ ỵ
(cA)
=
( c)A = 1A
=
A
c
c
ỏ s ❧➔

1
0mn = 0mn
c

✭✶✳✹✮

❚ø ✭✶✳✸✮ ✈➔ ✭✶✳✹✮ t❛ ❝â A = 0mn . ❱✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t ❝á♥ ❧↕✐ ❧➔ ❞➵
❞➔♥❣ ✈➔ ✤÷đ❝ ①❡♠ ♥❤÷ ❝→❝ ❜➔✐ t➟♣ ✈➲ ♥❤➔✳

❈→❝❤ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ♠❛ tr➟♥ ❜➟❝ ♥❤➜t
❳➨t ♣❤÷ì♥❣ tr➻♥❤ ♠❛ tr➟♥

cX + A = B,
✶✽

✭✶✳✺✮


tr♦♥❣ ✤â A, B ❧➔ 2 ♠❛ tr➟♥ ❦➼❝❤ ❝ï m × n. ❚❤➯♠ ✈➔♦ ❝↔ 2 ✈➳ ❝õ❛ ✭✶✳✺✮ −A t❛
✤÷đ❝

cX + A + (−A) = B − A ⇔ cX + 0mn = B − A ⇔ cX = B − A

✭✶✳✻✮

✲◆➳✉ c = 0, B = A✱ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ ❝â ✈ỉ sè ♥❣❤✐➺♠✳ ❚➟♣ ♥❣❤✐➺♠ ❝õ❛
♥â ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ♠❛ tr➟♥ ❦➼❝❤ ❝ï m × n.
✲◆➳✉ c = 0, B = A✱ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ ✈ỉ ♥❣❤✐➺♠✳
✲◆➳✉ c = 0✱♥❤➙♥ ❝↔ ừ ợ

1
c

t ữủ

1

1
1
(cX) = (B A) ⇔ X = (B − A)
c
c
c

✭✶✳✼✮

❱➼ ❞ö ✶✳✶✶✳ ❚➻♠ ♠❛ tr➟♥ X ✱ ❜✐➳t
✭✶✳✽✮

3X + A = B,
tr♦♥❣ ✤â

A=

●✐↔✐✿
− 34
2
3

1 −2
−3 4
,B =
0 3
2 1

❚❛ ❝â

X = 31 (B − A) = 31 (

−3 4
1 −2
−
)=
2 1
0 3

1
3

−4 6
=
2 −2

2
−2
3

✣à♥❤ ỵ t t ừ tr ❈❤♦ A, B, C ❧➔ ❝→❝ ♠❛
tr➟♥ ✈ỵ✐ ❝→❝ ❦➼❝❤ ❝ï s❛♦ ❝❤♦ tø♥❣ ♣❤➨♣ t♦→♥ s❛✉ ✤➙② ✤÷đ❝ ①→❝ ✤à♥❤✱ ❝❤♦

c

❧➔

♠ët sè t❤ü❝✳ ❑❤✐ ✤â t❛ ❝â✿
✶✳

A(BC) = (AB)C

✷✳

A(B + C) = AB + AC

✸✳

(A + B)C = AC + BC

✹✳

c(AB) = (cA)B = A(cB)

❚❛ ❝❤➾ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❝❤➜t ✷✳ ●✐↔ sû A = [aij ] ❧➔ ♠ët ♠❛ tr➟♥
❦➼❝❤ ❝ï m × n✱B = [bij ], C = [cij ] ❧➔ ❤❛✐ ♠❛ tr➟♥ ❦➼❝❤ ❝ï n × p✳ ❑❤✐ ✤â ❤➺ sè ð
✈à tr➼ ij ❝õ❛ A(B + C) ❧➔ nk=1 aik (bkj + ckj ) = nk=1 (aik bkj + aik ckj ), ❜➡♥❣ ❤➺
sè ð ✈à tr➼ ij ❝õ❛ AB + AC ✳ ❱➟② A(B + C) = AB + AC.
❈❤ù♥❣ ♠✐♥❤✳

❱✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t ❝á♥ ❧↕✐ ✤÷đ❝ ①❡♠ ♥❤÷ ❜➔✐ t➟♣ ✈➲ ♥❤➔✳
✶✾


ú ỵ tr õ t ❝❤➜t ❦➳t ❤ñ♣✱ ♥➯♥ t❛ ❝â t❤➸ ✈✐➳t
t➼❝❤ ♥❤✐➲✉ ♠❛ tr➟♥ ♠➔ ❦❤ỉ♥❣ ❝➛♥ ❞ị♥❣ ❞➜✉
✈✐➳t

()

✤➸ ♥❤â♠ ❝→❝ ♠❛ tr➟♥

A1 A2 ...Ak .

ú ỵ P tr➟♥ ❦❤æ♥❣ ❝â t➼♥❤ ❝❤➜t ❣✐❛♦ ❤♦→♥✳ ❚ù❝ ❧➔ ♥â✐
❝❤✉♥❣✱ t❛ ❦❤æ♥❣ ❝â

AB = BA✳

❚❤➟t ✈➟②✱ ❣✐↔ sû A ❧➔ ♠ët ♠❛ tr➟♥ ❦➼❝❤ ❝ï m × n✱ B ❧➔ ♠ët ♠❛ tr➟♥ ❦➼❝❤
❝ï n × p✳ ❚❛ ❝â AB ởt tr ù m ì p

ã p ì m t t BA ổ
ã p = m = n✱ ❦❤✐ ✤â AB ❧➔ ♠ët ♠❛ tr➟♥ ❦➼❝❤ ❝ï m × m✱ BA ❧➔ ♠ët
♠❛ tr➟♥ ❦➼❝❤ ❝ï n × n✱ ♥❤ú♥❣ ✈➻ m = n✱ ♥➯♥ AB = BA✳
• ◆❣❛② ❝↔ ❦❤✐ p = m = n✱ ❦❤✐ ✤â AB ✈➔ BA ✤➲✉ ①→❝ õ ũ
ù m ì m ữ ú ❝â t❤➸ ❦❤ỉ♥❣ ❜➡♥❣ ♥❤❛✉✱ ♥❤÷ tr♦♥❣ ✈➼ ❞ư ♠➔
❝❤ó♥❣ t❛ s➩ ✤÷❛ r❛ ♥❣❛② s❛✉ ✤➙②✳

❱➼ ❞ư ✶✳✶✷✳ ❈❤♦ A =

1 3
2 −1
,B =
.
2 −1
0 2

❑❤✐ ✤â

AB =

1 3
2 −1

2 −1
2 5
=
0 2
4 −4

BA =

2 −1
0 2

1 3
0 7
=
2 −1
4 −2

AB = BA.

❚❛ t❤➜② tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②

❦❤ỉ♥❣

◆❤÷♥❣ tø ✈➼ ❞ư ♥➔②✱ ❝❤ó♥❣ t❛ ❝ơ♥❣ ❦❤ỉ♥❣ t❤➸ ❦➳t ❧✉➟♥ r➡♥❣ AB
❜➡♥❣ BA✳ ❈â ♥❤ú♥❣ tr÷í♥❣ ❤đ♣ ♠➔ AB = BA ữ tr ử ữợ

ớ

ử A =

1 2
−2 4
,B =
.
1 1
2 −2

❚❛ ❝â✿

−2 4
2 0
=
2 −2
0 2

AB =

1 2
1 1

BA =

−2 4
2 −2

❚❛ t❤➜② tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✿

1 2
2 0
=
1 1
0 2

AB = BA.
✷✵


ú ỵ P tr ổ õ t ữợ tự tứ AC = BC
ổ ❧ó❝ ♥➔♦ ❝ơ♥❣ s✉② r❛

❱➼ ❞ư ✶✳✶✹✳ ❈❤♦ A =

A = B✳

1 3
2 4
1 −2
,B =
,C =
2 3
−1 2
0 1

❚❛ ❝â✿

❚❛ t❤➜②

AC = BC ✱

AC =

1 3
0 1

1 −2
−2 4
=
−1 2
−1 2

BC =

2 4
2 3

1 −2
−2 4
=
−1 2
−1 2

♥❤÷♥❣

A = B.

ú ỵ r t t t s t r➡♥❣ ♥➳✉ ❝â t❤➯♠ ✤✐➲✉ ❦✐➺♥ C
❦❤↔ ♥❣❤à❝❤✱ t❤➻ tø

AC = BC

s➩ s✉② r❛

A = B.

✣à♥❤ ♥❣❤➽❛ ✶✳✼ ✭▼❛ tr➟♥ ✤ì♥ ✈à✮✳ ▼❛ tr➟♥ ✤ì♥ ✈à ❝➜♣ n✱ ❦➼ ❤✐➺✉ ❧➔ I ✱ ❧➔
n

♠ët ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n ❝â ❝→❝ ❤➺ sè tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ❜➡♥❣ 1✱ t➜t ❝↔
❝→❝ ❤➺ sè ❝á♥ ❧↕✐ ❜➡♥❣ 0✳



1
0
In = 
...
0

❱➼ ❞ö ✶✳✶✺✳

0
1
...
0

...
...
...
...

0
0
...
0


0
0

...
1



1 0 0
1 0
I1 = 1 , I2 =
, I3 = 0 1 0
0 1
0 0 1

❑❤✐ ❝➜♣ ❝õ❛ ♠❛ tr➟♥ ✤ì♥ ✈à ✤➣ ❜✐➳t rã✱ t❛ ❝â t❤➸ ✈✐➳t I t In .

ỵ t t ❝õ❛ ♠❛ tr➟♥ ✤ì♥ ✈à✮✳ ❈❤♦ A ❧➔ ♠ët ♠❛ tr➟♥ ❦➼❝❤

❝ï

m × n✳

❑❤✐ ✤â t❛ ❝â✿

✶✳

AIn = A

✷✳

Im A = A

ự

ự ỵ ♥❤÷ ❜➔✐ t➟♣ ✈➲ ♥❤➔✳
✷✶


✣à♥❤ ♥❣❤➽❛ ✶✳✽ ✭▲ơ② t❤ø❛ ❝õ❛ ♠❛ tr➟♥ ✈✉ỉ♥❣✮✳ ❈❤♦ A ❧➔ ♠ët ♠❛ tr➟♥ ✈✉æ♥❣

❝➜♣ n✱ ❝❤♦ k ❧➔ ♠ët sè ♥❣✉②➯♥ ❦❤ỉ♥❣ ➙♠✳ ▲ơ② t❤ø❛ ❜➟❝ k ❝õ❛ A✱ ❦➼ ❤✐➺✉ ❧➔ Ak ✱
✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿
Ak = AA...A
k



q ữợ A0 = In

t Aj Ak = Aj+k , (Aj )k = Ajk ✈ỵ✐ ♠å✐ sè ♥❣✉②➯♥ ❦❤ỉ♥❣ ➙♠ j, k ✳

❱➼ ❞ư ✶✳✶✻✳ ❈❤♦ A =
●✐↔✐✿
1 −2
6 −3

❚❛

❝â

2 −1
.
3 0

A3

=

❍➣② t➻♠

AAA

A3 .

=

(

2 −1

3 0

2 −1
2 −1
)
3 0
3 0

=

2 −1
−4 −1
=
.
3 0
3 −6

✶✳✹✳ P❤➨♣ ❝❤✉②➸♥ ✈à ♠❛ tr➟♥
✣à♥❤ ♥❣❤➽❛ ✶✳✾ ✭▼❛ tr➟♥ ❝❤✉②➸♥ ✈à✮✳ ❈❤♦ A ❧➔ ♠ët ♠❛ tr➟♥ ❦➼❝❤ ❝ï m × n✳
▼❛ tr➟♥ ❝❤✉②➸♥ ✈à ❝õ❛ A✱ ❦➼ ❤✐➺✉ ❧➔ AT ✱ ởt tr ù n ì m
ữủ tø A ❜➡♥❣ ❝→❝❤ ❝❤✉②➸♥ ❞á♥❣ ❝õ❛ A t❤➔♥❤ ❝ët ❝õ❛ AT ✳




a11 a12 ... a1n
a11 a21 ... am1
 a21 a22 ... a2n 



 =⇒ AT =  a12 a22 ... am2 
A=
 ...
 ... ... ... ... 
... ... ... 
am1 am2 ... amn
a1n a2n ... amn

❱➼ ❞ö ✶✳✶✼✳ ❈❤♦
●✐↔✐✿ AT =



0 1
A = 2 4 ✳
1 −1

❚➻♠

AT ✳

0 2 1
1 4 1

ỵ t t ừ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à✮✳ ❈❤♦ A, B ❧➔ ❝→❝ ♠❛ tr
ợ ù s tứ t ữợ ✤➙② ①→❝ ✤à♥❤✱ ❝❤♦
❑❤✐ ✤â t❛ ❝â✿
✶✳

(AT )T = A

✷✳

(A + B)T = AT + B T
✷✷

c

❧➔ ♠ët sè t❤ü❝✳


✸✳

(cA)T = cAT

✹✳

(AB)T = B T AT

❚❛ ❝❤➾ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❝❤➜t ✹✳ ●✐↔ sû A = [aij ] ❧➔ ♠ët ♠❛
tr➟♥ ❦➼❝❤ ❝ï m × n✱ B = [bij ] ❧➔ ♠ët ♠❛ tr➟♥ ❦➼❝❤ ❝ï n × p✳ ❑❤✐ ✤â ✈ỵ✐ ♠é✐
i ∈ {1, ..., m}, j ∈ {1, ..., n} ❤➺ sè ð ✈à tr➼ ij ❝õ❛ ♠❛ tr➟♥ (AB)T ❜➡♥❣ ❤➺ sè ð
✈à tr➼ ji ❝õ❛ AB ✈➔ ❜➡♥❣ nk=1 ajk bki = nk=1 (B T )ik (AT )kj = (B T AT )ij , tr♦♥❣
✤â ✈ỵ✐ ♠é✐ ♠❛ tr➟♥ C ✱ Cij ❧➔ ❦➼ ❤✐➺✉ ❝❤♦ ❤➺ sè ð ✈à tr➼ ij ❝õ❛ ♠❛ tr➟♥ C ✳ ❱➟②
(AB)T = B T AT .
❈❤ù♥❣ ♠✐♥❤✳

❱✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t ❝á♥ ❧↕✐ ❞ü❛ ✈➔♦ ✤à♥❤ ♥❣❤➽❛ ✈➔ ①❡♠ ♥❤÷ ❜➔✐
t➟♣ ✈➲ ♥❤➔✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✵ ✭▼❛ tr➟♥ ✤è✐ ①ù♥❣✮✳ ▼❛ tr➟♥ ✈✉ỉ♥❣ A ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥
✤è✐ ①ù♥❣✱ ♥➳✉ A = A ✳
T

❱➼ ❞ö ✶✳✶✽✳ ❈❤♦



1 2 0
A = 2 −1 0✳
0 0 1

❚❛ ❝â



1 2 0
AT = 2 −1 0 = A✳
0 0 1

❱➟②

A

❧➔

♠ët ♠❛ tr ố ự

ỵ

A

ợ ộ tr

t ổ ❝â

AAT , AT A

❧➔ ❝→❝ ♠❛ tr➟♥

✤è✐ ①ù♥❣✳
✷✳ ❱ỵ✐ ♠é✐ ♠❛ tr➟♥ ✈✉æ♥❣

❈❤ù♥❣ ♠✐♥❤✳

A✱

t❛ ❧✉æ♥ ❝â

A + AT

❧➔ ♠ët ♠❛ tr➟♥ ✤è✐ ①ù♥❣✳

✶✳ ●✐↔ sû A ❝â ❦➼❝❤ ❝ï m × n✳ ❑❤✐ ✤â AT ❝â ❦➼❝❤ ❝ï n ×

m✳ ❉♦ ✤â AAT , AT A ①→❝ ✤à♥❤✳ ❚❛ t❤➜② (AAT )T
(AT )T AT

t t ừ ỵ
=

t t ừ ỵ
=

AAT

AAT ố ự ữỡ tü✱ AT A ✤è✐ ①ù♥❣✳
✷✳ ●✐↔ sû A ❧➔ ♠ët ♠❛ tr➟♥ ✈✉ỉ♥❣ ❝➜♣ n✳ ❑❤✐ ✤â AT ❝ơ♥❣ ❧➔ ♠ët ♠❛ tr➟♥
✈✉æ♥❣ ❝➜♣ n✳ ❉♦ ✤â A + AT

t t ừ ỵ

(A + AT )T
=
AT + (AT )T
A = A + AT ❉♦ ✤â A + AT ố ự



t t ừ ỵ ✶✳✺
=

AT +


✶✳✺✳ ▼❛ tr➟♥ ❞↕♥❣ ❜➟❝ t❤❛♥❣ t❤❡♦ ❞á♥❣ ✈➔ ♣❤➨♣ ❦❤û
●❛✉ss
✣à♥❤ ♥❣❤➽❛ ✶✳✶✶✳ ▼ët ♠❛ tr➟♥ ❞↕♥❣ ❜➟❝ t❤❛♥❣ t❤❡♦ ❞á♥❣ ❝â ❝→❝ t➼♥❤ ❝❤➜t
s❛✉✿

✶✳ ❈→❝ ❞á♥❣ ❝❤ù❛ t♦➔♥ 0 ữợ ũ ừ tr
ợ ộ ❞á♥❣ ❦❤æ♥❣ ❝❤ù❛ t♦➔♥ 0✱ ❤➺ sè ❦❤→❝ 0 ✤➛✉ t✐➯♥ ❝õ❛ ❞á♥❣ ✤â t➼♥❤
tø ❜➯♥ tr→✐ s❛♥❣ ❜➡♥❣ 1 ✈➔ ✤÷đ❝ ❣å✐ ❧➔ ❤➺ sè 1 ❞➝♥ ✤➛✉ ❝õ❛ ❞á♥❣ ✤â✳
✸✳ ✣è✐ ✈ỵ✐ 2 ❞á♥❣ ❦❤ỉ♥❣ ❝❤ù❛ t♦➔♥ 0✱ ❤➺ sè 1 ❞➝♥ ✤➛✉ ❝õ❛ ❞á♥❣ tr➯♥ ♥➡♠
❧➺❝❤ ✈➲ ❜➯♥ tr→✐ ♥❤✐➲✉ ❤ì♥ s♦ ✈ỵ✐ ❤➺ sè 1 ❞➝♥ ừ ỏ ữợ

ởt tr ❞↕♥❣ ❜➟❝ t❤❛♥❣ t❤❡♦ ❞á♥❣ ❝â t❤➯♠ t➼♥❤
❝❤➜t s❛✉✿

✲✣è✐ ✈ỵ✐ ❤➺ sè 1 ❞➝♥ ✤➛✉ ❝õ❛ ♠é✐ ❞á♥❣✿ t➜t ❝↔ số t ởt
tr ữợ ♥â ✤➲✉ ❜➡♥❣ 0✳
t❤➻ ♠❛ tr➟♥ ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ❞↕♥❣ ❜➟❝ t❤❛♥❣ t❤❡♦ ❞á♥❣ rót ❣å♥✳

❱➼ ❞ư ✶✳✶✾✳ ❈→❝ ♠❛ tr➟♥ s❛✉ ❝â ❞↕♥❣ ❜➟❝ t❤❛♥❣ t❤❡♦ ❞á♥❣
❛✮



1 2 −1 4
0 1 0
3
0 0 1 −2

❜✮



0 1 0 5
0 0 1 3
0 0 0 0

❝✮



1 −5 2 −1 3
0 0 1 3 −2


0 0 0 1
4
0 0 0 0
1

❞✮


1
0

0

0
0

0
1
0
0
0


0 −1
0 2

1 3

0 0
0 0
✷✹


❚r♦♥❣ ✤â✱ ❝→❝ ♠❛ tr➟♥ ð ♠ö❝ ❜✮✱ ❞✮ ❧➔ ❝â ❞↕♥❣ ❜➟❝ t❤❛♥❣ t❤❡♦ ❞á♥❣ rót ❣å♥✱
❝á♥ ❝→❝ ♠❛ tr➟♥ ð ♠ư❝ ❛✮✱❝✮ ❦❤ỉ♥❣ ❝â ❞↕♥❣ ❜➟❝ t❤❛♥❣ t❤❡♦ ❞á♥❣ rót ❣å♥✳

❱➼ ❞ư ✶✳✷✵✳ ❈→❝ ♠❛ tr➟♥ s❛✉ ❦❤ỉ♥❣ ❝â ❞↕♥❣ ❜➟❝ t❤❛♥❣ t❤❡♦ ❞á♥❣

❡✮


1 2 −3 4
0 2 1 −1
0 0 1 −3

✈➻ ❤➺ sè ❦❤→❝

0

✤➛✉ t✐➯♥ t➼♥❤ tø tr→✐ s❛♥❣ ❝õ❛ ❞á♥❣ t❤ù

❤❛✐ ❝õ❛ ♠❛ tr➟♥ ❦❤æ♥❣ ❜➡♥❣ ✶✳


❢✮


1 2 −1 2
0 0 0
0
0 1 2 −4

✈➻ ỏ ự t

0

ổ ữợ ũ ừ

tr

✤÷❛ ♠ët ♠❛ tr➟♥ ❜➜t ❦ý ✈➲ ♠❛ tr➟♥ ❞↕♥❣ ❜➟❝ t❤❛♥❣ t❤❡♦ ❞á♥❣ ❤♦➦❝
❞↕♥❣ ❜➟❝ t❤❛♥❣ t❤❡♦ ❞á♥❣ rót ❣å♥ ✭✤➸ t➻♠ ♠❛ tr➟♥ ♥❣❤à❝❤ ✤↔♦ ♥❤÷ tr♦♥❣ ♣❤➛♥
✶✳✻✳✱ ❤♦➦❝ ✤➸ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ♥❤÷ tr♦♥❣ ❈❤÷ì♥❣ ✷✮✱ t❛ s➩ ❞ị♥❣
❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ t❤✉ë❝ ❝→❝ ❞↕♥❣ s❛✉✿
✶✳ ✣ê✐ ❝❤é ✷ ❞á♥❣ ❝õ❛ ♠❛ tr➟♥✳
✷✳ ◆❤➙♥ ✶ ❞á♥❣ ❝õ❛ ♠❛ tr➟♥ ✈ỵ✐ ✶ sè t❤ü❝ ❦❤→❝ 0✳
✸✳ ❚❤➯♠ ✈➔♦ ✶ ❞á♥❣ ✶ sè ❧➛♥ ❝õ❛ ✶ ❞á♥❣ ❦❤→❝✳
◆❤ú♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ sì ❝➜♣ t❤❡♦ ❞á♥❣
✤è✐ ✈ỵ✐ ♠❛ tr➟♥✳ ❚❛ ♥â✐ ♠❛ tr➟♥ B t÷ì♥❣ ✤÷ì♥❣ t❤❡♦ ❞á♥❣ ợ tr A
B ữủ tứ A s ♠ët sè ❤ú✉ ❤↕♥ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ sì ❝➜♣ t❤❡♦ ❞á♥❣✳
◗✉→ tr➻♥❤ ❞ị♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ sì ❝➜♣ t❤❡♦ ❞á♥❣ ✤➸ ✤÷❛ ♠ët ♠❛ tr➟♥ ✈➲

❞↕♥❣ ❜➟❝ t❤❛♥❣ t❤❡♦ ❞á♥❣ ✭t÷ì♥❣ ù♥❣✱ ❞↕♥❣ ❜➟❝ t❤❛♥❣ t❤❡♦ ❞á♥❣ rót ❣å♥✮
✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ ❦❤û ●❛✉ss ✭t÷ì♥❣ ù♥❣✱ ♣❤➨♣ ❦❤û ●❛✉ss✲❏♦r❞❛♥ ✮✳ ❈→❝ ♣❤➨♣
❦❤û ♥➔② ✤÷đ❝ ✤➦t t❤❡♦ t➯♥ ❝õ❛ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ❈✳❋✳ ●❛✉ss ✭✶✼✼✼✲✶✽✺✺✮ ✈➔
❲✳ ❏♦r❞❛♥ ✭✶✽✹✷✲✶✽✾✾✮✳ ✣➸ t❤✉➟♥ t✐➺♥✱ t❛ sû ❞ö♥❣ ❝→❝ ❦➼ ❤✐➺✉ s❛✉ ❝❤♦ ❝→❝ ♣❤➨♣
❜✐➳♥ ✤ê✐ sì ❝➜♣ t❤❡♦ ❞á♥❣ ✤è✐ ✈ỵ✐ ♠❛ tr➟♥✿

• ❉ị♥❣ ❦➼ ❤✐➺✉ Ri ↔ Rj ❦❤✐ t❛ ✤ê✐ ❝❤é ❞á♥❣ t❤ù i ✈➔ ❞á♥❣ t❤ù j ✱ ð ✤➙② R
❧➔ ✈✐➳t t➢t ❝õ❛ tø t✐➳♥❣ ❆♥❤ ✧r♦✇✧ ✭❞á♥❣✮✳
• ❉ị♥❣ ❦➼ ❤✐➺✉ αRi → Ri ❦❤✐ t❛ ♥❤➙♥ ❞á♥❣ t❤ù i ✈ỵ✐ sè t❤ü❝ α = 0✳
✷✺