ÁLGEBRA LINEAR: UMA CONEXÃO DO ENSINO MÉDIO AO SUPERIOR

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚

➪▲●❊❇❘❆ ▲■◆❊❆❘✿ ❯▼❆

❈❖◆❊❳➹❖ ❉❖ ❊◆❙■◆❖ ▼➱❉■❖

❆❖ ❙❯P❊❘■❖❘

♣♦r

❍➪▲■❙❙❖◆ ❇❆❘❘❊❚❖ ❱■❊■❘❆

s♦❜ ♦r✐❡♥t❛çã♦ ❞❛

Pr♦❢✳ ❉r✳ ❏♦ã♦ ▼❛r❝♦s ❇❡③❡rr❛ ❞♦ Ó

❡ ❝♦♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢❛✳ ▼❡✳ ❋❧á✈✐❛ ❏❡rô♥✐♠♦ ❇❛r❜♦s❛

❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥✲ t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✲ ✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦✲ ♥❛❧ ✲ P❘❖❋▼❆❚ ✲ ❈❈❊◆ ✲ ❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❆❣♦st♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇

❆❣r❛❞❡❝✐♠❡♥t♦s

❉❡❞✐❝❛tór✐❛

❘❡s✉♠♦

❊st❡ tr❛❜❛❧❤♦ é ✉♠ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s s♦❜ ✉♠❛ ♣❡rs♣❡❝t✐✈❛ ❞❛ ➪❧❣❡❜r❛ ▲✐♥❡❛r✳ ❯t✐❧✐③❛r❡♠♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ ♠❛tr✐③✱ ✈❡t♦r✱ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r✱ ❞❡♣❡♥❞ê♥✲ ❝✐❛ ❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❧✐♥❡❛r✱ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ❜❛s❡ ❡ ❞✐♠❡♥sã♦✳ ❋❛r❡♠♦s t❛♠❜é♠ ♦ ❝á❧❝✉❧♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s ❡ ✐♠♣❧✐❝❛çõ❡s✳ ◆♦ss♦ ✐♥t✉✐t♦ é ❛♣r❡s❡♥t❛r ♦s r✉❞✐♠❡♥t♦s ❞❛ á❧❣❡❜r❛ ❧✐♥❡❛r ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ❛✉①✐❧✐❛❞♦r❛ ♥❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ❡ ❡①✐❜✐r ❛ s✉❛ ❣❡♦♠❡tr✐❛✳ ◗✉❡r❡♠♦s ❝♦♠ ✐st♦ ❝♦♥❢❡❝❝✐♦♥❛r ✉♠ t❡①t♦ ❛✉①✐❧✐❛r q✉❡ ♣♦ss❛ s❡r ❡①♣❧♦r❛❞♦ ♣♦r ❡st✉❞❛♥t❡s ❡ ♣r♦❢❡ss♦r❡s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❡✱ ❛ss✐♠✱ s✉❛✈❡♠❡♥t❡ ✐♥tr♦❞✉③✐♥❞♦ ❡st❛ ♣♦❞❡r♦s❛ ❢❡rr❛♠❡♥t❛ ❞❛ ♠❛t❡♠át✐❝❛✳ ◆♦ ❞❡❝♦rr❡r ❞♦ t❡①t♦ s❡rã♦ ❛❜♦r❞❛❞♦s t❛♠❜é♠ ❛❧❣✉♥s ❛s♣❡❝t♦s ❤✐stór✐❝♦s✳

P❛❧❛✈r❛s ❝❤❛✈❡s✿ ▼❛tr✐③❡s✱ ❉❡t❡r♠✐♥❛♥t❡s✱ ❙✐st❡♠❛s ▲✐♥❡❛r❡s✱ ➪❧❣❡❜r❛ ▲✐♥❡❛r✱ ❊♥s✐♥♦ ▼é❞✐♦✳

❆❜str❛❝t

❚❤✐s ✇♦r❦ ✐s ❛ st✉❞② ♦❢ ❧✐♥❡❛r s②st❡♠s ❢r♦♠ t❤❡ ♣❡rs♣❡❝t✐✈❡ ♦❢ ❧✐♥❡❛r ❛❧❣❡❜r❛✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❝♦♥❝❡♣ts ♦❢ ♠❛tr✐①✱ ✈❡❝t♦r✱ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥✱ ❧✐♥❡❛r ❞❡♣❡♥❞❡♥❝❡ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥❝❡✱ ✈❡❝t♦r s♣❛❝❡✱ ❜❛s✐s ❛♥❞ ❞✐♠❡♥s✐♦♥✳ ❲❡ ✇✐❧❧ ❛❧s♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❞❡t❡r♠✐♥❛♥ts ❛♥❞ ✐♠♣❧✐❝❛t✐♦♥s✳ ❖✉r ❛✐♠ ✐s t♦ ♣r❡s❡♥t t❤❡ r✉❞✐♠❡♥ts ♦❢ ▲✐♥❡❛r ❆❧✲ ❣❡❜r❛ ❛s ❤❡❧♣❡r t♦♦❧ ✐♥ s♦❧✈✐♥❣ ❧✐♥❡❛r s②st❡♠s ❛♥❞ ❞✐s♣❧❛② ✐ts ❣❡♦♠❡tr②✳ ❲❡ ✇❛♥t ✐t t♦ ♠❛♥✉❢❛❝t✉r❡ ❛ ❛✉①✐❧✐❛r② t❡①t t❤❛t ❝❛♥ ❜❡ ❡①♣❧♦r❡❞ ❜② st✉❞❡♥ts ❛♥❞ ❤✐❣❤ s❝❤♦♦❧ t❡❛❝❤❡rs✱ ❛♥❞ s♦ ❣❡♥t❧② ✐♥tr♦❞✉❝✐♥❣ t❤✐s ♣♦✇❡r❢✉❧ ♠❛t❤❡♠❛t✐❝❛❧ t♦♦❧✳ ❚❤r♦✉❣❤♦✉t t❤❡ t❡①t ✇✐❧❧ ❜❡ ❝♦✈❡r❡❞ ❛❧s♦ s♦♠❡ ❤✐st♦r✐❝❛❧ ❛s♣❡❝ts✳

❑❡②✇♦r❞s✿ ▼❛tr✐❝❡s✱ ❉❡t❡r♠✐♥❛♥ts✱ ▲✐♥❡❛r ❙②st❡♠s✱ ▲✐♥❡❛r ❆❧❣❡❜r❛✱ ❍✐❣❤ ❙❝❤♦♦❧✳

❙✉♠ár✐♦

✶ ▼❆❚❘■❩❊❙ ✶

✶✳✶ ❆s♣❡❝t♦s ❤✐stór✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❉❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ❚✐♣♦s ❞❡ ♠❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸✳✶ ▼❛tr✐③ q✉❛❞r❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸✳✷ ▼❛tr✐③ tr✐❛♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸✳✸ ▼❛tr✐③ ❞✐❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸✳✹ ▼❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸✳✺ ▼❛tr✐③ ♥✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸✳✻ ▼❛tr✐③ tr❛♥s♣♦st❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸✳✼ ▼❛tr✐③ s✐♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✹ ■❣✉❛❧❞❛❞❡ ❞❡ ♠❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✺ ❖♣❡r❛çõ❡s ❝♦♠ ♠❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✺✳✶ ❆❞✐çã♦ ❞❡ ♠❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✺✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠ ❡s❝❛❧❛r ♣♦r ✉♠❛ ♠❛tr✐③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✺✳✸ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✺✳✹ ▼❛tr✐③❡s ✐♥✈❡rtí✈❡✐s ✭♦✉ ✐♥✈❡rsí✈❡✐s✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✷ ❉❊❚❊❘▼■◆❆◆❚❊❙ ✶✷

✷✳✶ ❆s♣❡❝t♦s ❤✐stór✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✷ ❉❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✸ ❉❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ tr✐❛♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✹ Pr♦♣r✐❡❞❛❞❡s ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✸ ❙■❙❚❊▼❆❙ ▲■◆❊❆❘❊❙ ✶✻

✸✳✶ ❆s♣❡❝t♦s ❤✐stór✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✷ ❉❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✸ ❋♦r♠❛ ♠❛tr✐❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✹ ❙✐st❡♠❛ ▲✐♥❡❛r ❍♦♠♦❣ê♥❡♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✺ ❙✐st❡♠❛ ▲✐♥❡❛r ❊q✉✐✈❛❧❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✻ ❈❧❛ss✐✜❝❛çã♦ ❞❡ ✉♠ ❙✐st❡♠❛ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✸✳✼ ❘❡s♦❧✉çã♦ ❞❡ ✉♠ ❙✐st❡♠❛ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✼✳✶ ❊❧✐♠✐♥❛çã♦ ❣❛✉ss✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✹ ❱❊❚❖❘❊❙ ❊ ➪▲●❊❇❘❆ ▲■◆❊❆❘ ✷✷

✹✳✶ ❆s♣❡❝t♦s ❤✐stór✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✹✳✷ ❉❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✸ ❱❡t♦r❡s ❡q✉✐♣♦❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✹ ❯s♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❞❡ ✉♠ ✈❡t♦r ♥♦ ♣❧❛♥♦ R2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✹✳✹✳✶ ■❣✉❛❧❞❛❞❡ ❞❡ ✈❡t♦r❡s ♥♦ R2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✹✳✷ ❖♣❡r❛çõ❡s ❝♦♠ ✈❡t♦r❡s ♥♦ R2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✹✳✹✳✸ ❈♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ♥♦ R2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✹✳✹ ❉❡♣❡♥❞ê♥❝✐❛ ❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❧✐♥❡❛r ♥♦ R2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✺ ❯s♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❞❡ ✉♠ ✈❡t♦r ♥♦ ❡s♣❛ç♦ R3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✹✳✺✳✶ ■❣✉❛❧❞❛❞❡ ❞❡ ✈❡t♦r❡s ♥♦ R3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✺✳✷ ❖♣❡r❛çõ❡s ❝♦♠ ✈❡t♦r❡s ♥♦ R3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✺✳✸ ❈♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ♥♦ R3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✹✳✺✳✹ ❉❡♣❡♥❞ê♥❝✐❛ ❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❧✐♥❡❛r ♥♦ R3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✻ ➪❧❣❡❜r❛ ❧✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✻✳✶ ❊s♣❛ç♦s ✈❡t♦r✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✻✳✷ ❙✉❜❡s♣❛ç♦s ✈❡t♦r✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✻✳✸ ❈♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✻✳✹ ❉❡♣❡♥❞ê♥❝✐❛ ❡ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❧✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✻✳✺ ❇❛s❡ ❡ ❞✐♠❡♥sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✺ ❆P▲■❈❆➬Õ❊❙ ◆❖ ❊◆❙■◆❖ ▼➱❉■❖ ✺✺

✺✳✶ ✶❛ _{❛♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺}

✺✳✷ ✷❛ _{❛♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼}

✺✳✸ ✸❛ _{❛♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷}

✺✳✹ ✹❛ _{❛♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹}

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✼✵

■♥tr♦❞✉çã♦

❊st❛ ❞✐ss❡rt❛çã♦ tr❛t❛ ❞❡ ✉♠❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ t❡①t♦ ❝♦♠♣❧❡♠❡♥t❛r ♣❛r❛ ♦ ❡st✉❞♦ ❞♦s s✐st❡♠❛s ❧✐♥❡❛r❡s✳ ❖ ♣ú❜❧✐❝♦ ❛❧✈♦ sã♦ ♦s ♣r♦❢❡ss♦r❡s ❡ ❡st✉❞❛♥t❡s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❖ ♣r♦♣ós✐t♦ ❛q✉✐ é ❛♣r❡s❡♥t❛r ♦s s✐st❡♠❛s ❧✐♥❡❛r❡s s♦❜ ❛ ót✐❝❛ ❞❛ ➪❧❣❡❜r❛ ▲✐♥❡❛r✳ ◆♦ss♦ ♦❜❥❡t✐✈♦ é ♣r♦♣♦r❝✐♦♥❛r ✉♠❛ ✈✐sã♦ ♣❛♥♦râ♠✐❝❛ ❞❡st❡ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛ ❡ ❢❛③❡r ❝♦♠ q✉❡ ♦ ❧❡✐t♦r ❡♥t❡♥❞❛ q✉❡ ♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s s❡ ❢✉♥❞❡ ❝♦♠ ♦ ❞❛s ♠❛tr✐③❡s ❡✱ ❡st❡✱ ❝♦♠ ♦ ❞❡ ✈❡t♦r❡s✱ q✉❡ sã♦ ❡❧❡♠❡♥t♦s ❞❡ ❊s♣❛ç♦s ❱❡t♦r✐❛✐s✳

❆ ❡s❝♦❧❤❛ ❞♦ t❡♠❛ ❡♠ q✉❡stã♦ s❡ ❞❡✉ ❞❡✈✐❞♦ ❛ ❞✐✜❝✉❧❞❛❞❡ ❡①✐st❡♥t❡ ♥❛ ❛♣r❡♥❞✐③❛✲ ❣❡♠ ❞❛ ❞✐s❝✐♣❧✐♥❛✳ ❆t✉❛❧♠❡♥t❡✱ ♣r❡❝✐s❛♠♦s ❞❡ ❝r✐❛t✐✈✐❞❛❞❡ ❡ ❞✐s♣♦s✐çã♦ ♣❛r❛ ♠✉❞❛r ❛ ❢♦r♠❛ ❞❡ ❡♥s✐♥❛r ♠❛t❡♠át✐❝❛✱ s❛✐r ✉♠ ♣♦✉❝♦ ❞♦ tr❛❞✐❝✐♦♥❛❧✐s♠♦ ❡ ❢❛③❡r ❝♦♠ q✉❡ ♦ ❛❧✉♥♦ t❡♥❤❛ ✉♠❛ ✈✐sã♦ ♠❛✐s ❛♠♣❧❛ ❞❛ ♠❛t❡♠át✐❝❛✱ ❝♦♠♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ q✉❡ ❡stá ♣r❡s❡♥t❡ ❡♠ ♠✉✐t❛s s✐t✉❛çõ❡s ❞♦ ♥♦ss♦ ❝♦t✐❞✐❛♥♦ ❡ q✉❡ s❡ ✐♥t❡r✲r❡❧❛❝✐♦♥❛ ❝♦♠ ❞✐✲ ✈❡rs❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ♥❛ ❊♥❣❡♥❤❛r✐❛✱ ❊st❛tíst✐❝❛✱ ❡♥tr❡ ♦✉tr❛s✳

❖ tr❛❜❛❧❤♦ ❢♦✐ ❞✐✈✐❞✐❞♦ ❡♠ ❝✐♥❝♦ ❝❛♣ít✉❧♦s✳ ❯♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❡ ❞❡ ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ❢♦r❛♠ ❛❜♦r❞❛❞♦s ❞❡ ❢♦r♠❛s s✐♠♣❧❡s✱ tr❛③❡♥❞♦ s✉❛✈✐❞❛❞❡ ❛ ❧❡✐t✉r❛ ❡✱ ❛ss✐♠✱ ❝❛t✐✈❛♥❞♦ ♦ ❧❡✐t♦r✱ ♣❛r❛ q✉❡ ❡st❡s✱ ♣♦ss❛♠ t❡r ✉♠❛ ♠❛✐♦r ✐♥t✐♠✐❞❛❞❡ ❝♦♠ ♦ ❛ss✉♥t♦✳

❖ ❝❛♣ít✉❧♦ ✶ ❛❜♦r❞❛ ♦s ❝♦♥❝❡✐t♦s ❞❡ ▼❛tr✐③❡s✱ ♦s t✐♣♦s✱ ❛s ♦♣❡r❛çõ❡s✿ ❞❡ ❛❞✐çã♦✱ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦r ✉♠❛ ♠❛tr✐③✱ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s❀ ❡ ♠❛tr✐③ ✐♥✈❡rs❛✳ ❆✐♥❞❛✱ ❢❛r❡♠♦s ♠❡♥çã♦ ❛ ❛❧❣✉♠❛s ❞❛s s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳

❖ ❝❛♣ít✉❧♦ ✷ ♠♦str❛ ✉♠ ♣♦✉❝♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s✱ ❡ ❛✐♥❞❛ r❡❧❡♠❜r❛ ❛❧❣✉♠❛s ❞❛s s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳

❖ ❝❛♣ít✉❧♦ ✸ ❞❛r❡♠♦s r❡❢❡rê♥❝✐❛ ❛♦ ❡st✉❞♦ ❞♦s s✐st❡♠❛s ❧✐♥❡❛r❡s✱ ❞❡ s✉❛ r❡♣r❡✲ s❡♥t❛çã♦ ♥❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧ ❡ ♥❛ s✉❛ r❡s♦❧✉çã♦✱ ✉t✐❧✐③❛♥❞♦ ❛ ❡❧✐♠✐♥❛çã♦ ❣❛✉ss✐❛♥❛✳

◆♦ ❝❛♣ít✉❧♦ ✹ ❛❜♦r❞❛r❡♠♦s ✈❡t♦r❡s✱ ❛s s✉❛s ♦♣❡r❛çõ❡s✿ ❞❡ ❛❞✐çã♦ ❡♥tr❡ ✈❡t♦r❡s✱ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r❀ ❡ ❛✐♥❞❛ ✉♠ ♣♦✉❝♦ ❞❡ á❧❣❡❜r❛ ❧✐♥❡❛r✿ ❡s♣❛ç♦s ❡ s✉❜❡s♣❛✲ ç♦s ✈❡t♦r✐❛✐s✱ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r✱ ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡s✱ ❜❛s❡ ❡ ❞✐♠❡♥sã♦✳

❖ ❝❛♣ít✉❧♦ ✺ ♠♦str❛ ♥✉♠❛ ♣❡rs♣❡❝t✐✈❛ ❞❛ ➪❧❣❡❜r❛ ▲✐♥❡❛r✱ ❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s✱ ✉t✐❧✐③❛♥❞♦ ♣❛r❛ ✐ss♦✱ ✈❡t♦r❡s✱ ♠❛tr✐③❡s ❡ s✉❜❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳

❈❛♣ít✉❧♦ ✶

▼❆❚❘■❩❊❙

❆♣r❡s❡♥t❛♠♦s ♥❡st❡ ❝❛♣ít✉❧♦ ❛s ♣r✐♥❝✐♣❛✐s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s s♦❜r❡ ♠❛tr✐③❡s✱ q✉❡ s❡rã♦ ♥❡❝❡ssár✐❛s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s✳

✶✳✶ ❆s♣❡❝t♦s ❤✐stór✐❝♦s

❍✐st♦r✐❝❛♠❡♥t❡✱ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❝♦♥❥✉♥t♦s ❞❡ ♥ú♠❡r♦s ❡♠ ❢♦r♠❛ ❞❡ ♠❛tr✐③❡s s✉r❣❡ ♥♦ sé❝✉❧♦ ❳■❳✱ ♣♦ré♠ ❤á ✈❡stí❣✐♦s ❞❡ q✉❡ ❞❡s❞❡ ❛ é♣♦❝❛ ❞❡ ✷✺✵✵ ❛✳ ❈✳ ♦s ❝❤✐♥❡s❡s ❥á s♦❧✉❝✐♦♥❛ss❡♠ ❛❧❣✉♥s t✐♣♦s ❞❡ ♣r♦❜❧❡♠❛s ❝♦♠ ❝á❧❝✉❧♦s ❡❢❡t✉❛❞♦s s♦❜r❡ ✉♠❛ t❛❜❡❧❛✳ ❊♠ ✶✽✷✻✱ ♦ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês ❆✉❣✉st✐♥✲▲♦✉✐s ❈❛✉❝❤② ❞❡♥♦♠✐♥♦✉ ❡ss❛s ❝♦♥✜❣✉r❛çõ❡s ♥✉♠ér✐❝❛s ❞❡ t❛❜❧❡❛✉ ✭q✉❡ s✐❣♥✐✜❝❛ t❛❜❡❧❛✱ ❡♠ ❢r❛♥❝ês✮✳ ▼❛s✱ s♦♠❡♥t❡ ❡♠ ✶✽✺✵ é q✉❡ ♦ ♠❛t❡♠át✐❝♦ ✐♥❣❧ês✱ ❏❛♠❡s ❏♦s❡♣❤ ❙②❧✈❡st❡r✱ ❞❡♥♦♠✐♥♦✉ ❡ss❡ t✐♣♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ♥✉♠ér✐❝❛ ❞❡ ♠❛tr✐③ ✭✈❡r ❡♠ ❬✹❪✮✳

❆ ❞✐s♣✉t❛ ❡♥tr❡ ◆❡✇t♦♥✱ ▲❡✐❜♥✐③ ❡ s❡✉s ❛❞❡♣t♦s✱ ❡♠ t♦r♥♦ ❞❛ ♣r✐♠❛③✐❛ ❞❛ ❝r✐✲ ❛çã♦ ❞♦ ❈á❧❝✉❧♦✱ ❢♦✐ ♥❡❣❛t✐✈❛ ♣❛r❛ ❛ ♠❛t❡♠át✐❝❛ ✐♥❣❧❡s❛✱ ❡♠❜♦r❛ ◆❡✇t♦♥ t✐✈❡ss❡ ❧❡✈❛❞♦ ✈❛♥t❛❣❡♠ ♥❡ss❛ ♣♦❧ê♠✐❝❛✳ ❈♦♥s✐❞❡r❛♥❞♦ ✉♠❛ q✉❡stã♦ ❞❡ ❤♦♥r❛ ♥❛❝✐♦♥❛❧ s❡r ✜❡❧ ❛♦ s❡✉ ♠❛✐s ❡♠✐♥❡♥t❡ ❝✐❡♥t✐st❛✱ ♥♦s ✶✵✵ ❛♥♦s s❡❣✉✐♥t❡s ❛♦ ✐♥í❝✐♦ ❞❡ss❡ ❡♣✐só❞✐♦ ♦s ♠❛t❡♠át✐❝♦s ❜r✐tâ♥✐❝♦s ✜①❛r❛♠✲s❡ ♥♦s ♠ét♦❞♦s ❣❡♦♠étr✐❝♦s ♣✉r♦s✱ ♣r❡❢❡r✐❞♦s ❞❡ ◆❡✇t♦♥✱ ❡♠ ❞❡tr✐♠❡♥t♦ ❞♦s ♠ét♦❞♦s ❛♥❛❧ít✐❝♦s✳ ❈♦♠♦ ♦s ♠❛t❡♠át✐❝♦s ❞❛ ❊✉r♦♣❛ ❈♦♥t✐♥❡♥t❛❧ ❡①♣❧♦r❛r❛♠ ❢♦rt❡♠❡♥t❡ ❡st❡s ú❧t✐♠♦s ♠ét♦❞♦s ♥❡ss❡ ♣❡rí♦❞♦✱ ❛ ♠❛t❡♠á✲ t✐❝❛ ❜r✐tâ♥✐❝❛✱ ❛❝❛❜♦✉ ✜❝❛♥❞♦ ♠❛r❣✐♥❛❧✐③❛❞❛✳ P♦ré♠✱ ❤♦✉✈❡ ✉♠❛ r❡❛çã♦ ❡ ❛ á❧❣❡❜r❛ ❢♦✐ ❞✉r❛♥t❡ ❛❧❣✉♠ t❡♠♣♦ q✉❛s❡ ✉♠ ♠♦♥♦♣ó❧✐♦ ❜r✐tâ♥✐❝♦✳ ❉❡♥tr❡ ♦s ♠❛✐♦r❡s r❡s♣♦♥✲ sá✈❡✐s ♣♦r ❡ss❛ r❡❛s❝❡♥sã♦ ❢♦✐ ❆rt❤✉r ❈❛②❧❡②✱ q✉❡ ❝♦♥tr✐❜✉✐✉ t❛♥t♦ ♣❛r❛ ❛ á❧❣❡❜r❛ q✉❛♥t♦ ♣❛r❛ ❛ ❣❡♦♠❡tr✐❛ ✭✈❡r ❡♠ ❬✼❪✮✳

❖ ✐♥í❝✐♦ ❞❛ t❡♦r✐❛ ❞❛s ♠❛tr✐③❡s r❡♠♦♥t❛ ❛ ✉♠ ❛rt✐❣♦ ♦♥❞❡ ❈❛②❧❡② ❢❡③ q✉❡stã♦ ❞❡ s❛❧✐❡♥t❛r q✉❡✱ ❡♠❜♦r❛ ♣❡❧❛ ❧ó❣✐❝❛ ❛ ✐❞❡✐❛ ❞❡ ♠❛tr✐③ ♣r❡❝❡❞❛ ❛ ❞❡ ❞❡t❡r♠✐♥❛♥t❡✱ ❤✐st♦r✐❝❛♠❡♥t❡ ♦❝♦rr❡✉ ♦ ✐♥✈❡rs♦✿ ❞❡ ❢❛t♦✱ ♦s ❞❡t❡r♠✐♥❛♥t❡s ❥á ❡r❛♠ ✉s❛❞♦s ❤á ❜❛s✲ t❛♥t❡ t❡♠♣♦ ♥❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s✳ ◆♦ q✉❡ s❡ r❡❢❡r❡ às ♠❛tr✐③❡s✱ ❈❛②❧❡②

▼❛tr✐③❡s ❈❛♣ít✉❧♦ ✶

✐♥tr♦❞✉③✐✉✲❛s ♣❛r❛ t♦r♥❛r ♠❛✐s s✐♠♣❧❡s ❛ ♥♦t❛çã♦ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r✳ ❊♥✲ tã♦✱ ❡♠ ❧✉❣❛r ❞❡

x′

=ax+by y′

=cx+dy ✉s❛✈❛ (x

′

, y′

) =

a b c d

(x, y)

❆♦ ♦❜s❡r✈❛r ♦ ❡❢❡✐t♦ ❞❡ ❞✉❛s tr❛♥s❢♦r♠❛çõ❡s s✉❝❡ss✐✈❛s ❈❛②❧❡② ❝♦♥❝❧✉✐✉ q✉❡ ❝❤❡✲ ❣❛r✐❛ à ❞❡✜♥✐çã♦ ❞❡ ♣r♦❞✉t♦ ❞❡ ♠❛tr✐③❡s✳ ◆❛ s❡q✉ê♥❝✐❛✱ ❝❤❡❣♦✉ ❛ ✐❞❡✐❛ ❞❡ ✐♥✈❡rs❛ ❞❡ ✉♠❛ ♠❛tr✐③✱ ♦ q✉❡ ♦❜✈✐❛♠❡♥t❡ ♣r❡ss✉♣õ❡ ❛ ❞❡ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ✭♥♦ ❝❛s♦✱ ❛ ♠❛✲ tr✐③ ✐❞❡♥t✐❞❛❞❡✮✳ ❚rês ❛♥♦s ❞❡♣♦✐s✱ ❡♠ ✉♠ ♦✉tr♦ ❛rt✐❣♦✱ é q✉❡ ❈❛②❧❡② ✐♥tr♦❞✉③✐✉ ♦s ❝♦♥❝❡✐t♦s ❞❡ ❛❞✐çã♦ ❞❡ ♠❛tr✐③❡s ❡ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s ♣♦r ❡s❝❛❧❛r❡s✱ ❞❛♥❞♦ ê♥❢❛s❡ ✐♥❝❧✉s✐✈❡ ♣❛r❛ ❛s ♣r♦♣r✐❡❞❛❞❡s ❛❧❣é❜r✐❝❛s ❞❡ss❛s ♦♣❡r❛çõ❡s ✭✈❡r ❡♠ ❬✼❪✮✳

❆♦ ❞❡s❡♥✈♦❧✈❡r t❛♥t♦ ❛ t❡♦r✐❛ ❞❛s ♠❛tr✐③❡s✱ ❝♦♠♦ ♦✉tr♦s ❛ss✉♥t♦s✱ ❛ ♠❛✐♦r ♣r❡✲ ♦❝✉♣❛çã♦ ❞❡ ❈❛②❧❡② ❡r❛ ❝♦♠ ❛ ❢♦r♠❛ ❡ ❛ ❡str✉t✉r❛ ❡♠ á❧❣❡❜r❛✳ ❖ sé❝✉❧♦ ❳❳ s❡ ❡♥❝❛rr❡❣❛r✐❛ ❞❡ ❡♥❝♦♥tr❛r ✐♥ú♠❡r❛s ❛♣❧✐❝❛çõ❡s ♣❛r❛ s✉❛s ♠❛tr✐③❡s ✭✈❡r ❡♠ ❬✼❪✮✳

✶✳✷ ❉❡✜♥✐çã♦

❯♠❛ ♠❛tr✐③ ❞♦ t✐♣♦ m_×n✱ ❝♦♠m ❡n♥ú♠❡r♦s ✐♥t❡✐r♦s ❡ ♣♦s✐t✐✈♦s✱ é ✉♠❛ t❛❜❡❧❛ r❡t❛♥❣✉❧❛r ❞❡ ♥ú♠❡r♦s r❡❛✐s ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ ❞✐s♣♦st♦s ❡♠ m ❧✐♥❤❛s ❡n ❝♦❧✉♥❛s✳ ❘❡♣r❡s❡♥t❛♠♦s ✉♠❛ ♠❛tr✐③ ♣♦r ✉♠❛ ❧❡tr❛ ♠❛✐ús❝✉❧❛ ❡ ✐♥❢♦r♠❛♠♦s ♦ s❡✉ t✐♣♦ ❡s❝r❡✲ ✈❡♥❞♦ ♣r✐♠❡✐r♦ ♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❡✱ ❡♠ s❡❣✉✐❞❛✱ ♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s✱ ♣♦r ❡①❡♠♣❧♦✿

A =

1 3 0

−2 1 5

é ✉♠❛ ♠❛tr✐③ ❞♦ t✐♣♦ 2_×3✳

B =





1 3 0

−2 1 5 0 1 3



 é ✉♠❛ ♠❛tr✐③ ❞♦ t✐♣♦ 3×3✳

C =

u1 v1 u2 v2 u3 v3

é ✉♠❛ ♠❛tr✐③ ❞♦ t✐♣♦ 3_×2✳

❈♦♠♦ ❢♦✐ ❞✐t♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ✉♠❛ ♠❛tr✐③ é ✉♠❛ ❧✐st❛ ❞❡ ♥ú♠❡r♦s aij✱ ❝♦♠ í♥❞✐❝❡s ❞✉♣❧♦s✱ ♦♥❞❡ 1_≤i_≤m ❡ 1_≤j _≤n✳

❯♠❛ ♠❛tr✐③ A✱ ❞♦ t✐♣♦ m_×n✱ é ✉♠❛ t❛❜❡❧❛ ♥❛ q✉❛❧ ♦ ❡❧❡♠❡♥t♦ aij ❡♥❝♦♥tr❛✲s❡ ♥♦ ❝r✉③❛♠❡♥t♦ ❞❛ i✲és✐♠❛ ❧✐♥❤❛ ✭♦✉ i✲és✐♠♦ ✈❡t♦r ❧✐♥❤❛✮ ❝♦♠ ❛ j✲és✐♠❛ ❝♦❧✉♥❛ ✭♦✉ j✲és✐♠♦ ✈❡t♦r ❝♦❧✉♥❛✮✱ ♦✉ s❡❥❛✿

▼❛tr✐③❡s ❈❛♣ít✉❧♦ ✶

Am×n =



   

a11 a12 · · · a1_n a21 a22 · · · a2_n ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ am1 am2 · · · amn

     .

❘❡♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ♠❛tr✐③ ❣❡♥ér✐❝❛ ♣♦r A = [aij]m×n✱ ❝♦♠ 1 ≤ i ≤ m ❡

1_≤j _≤n✱ ♦♥❞❡ aij s❡rá ❝❤❛♠❛❞♦ ❞❡ ❡❧❡♠❡♥t♦ ❣❡♥ér✐❝♦ ❞❛ ♠❛tr✐③ A✳

➱ ✐♠♣♦rt❛♥t❡ ♦❜s❡r✈❛r q✉❡ ❛ j✲és✐♠❛ ❝♦❧✉♥❛ ❞❛ ♠❛tr✐③ ❆✱ q✉❡ é ✉♠❛ ♠❛tr✐③ m_×1✱ s❡rá ❝❤❛♠❛❞❛ ❞❡ ✈❡t♦r j✲és✐♠♦✱ ❡ ❞❡♥♦t❛r❡♠♦s ♣♦r vj~✱ ♦✉ s❡❥❛✿

~ vj =     

a1_j a2j ✳✳✳ amj      .

✶✳✸ ❚✐♣♦s ❞❡ ♠❛tr✐③❡s

✶✳✸✳✶ ▼❛tr✐③ q✉❛❞r❛❞❛

❯♠❛ ♠❛tr✐③ Qn s❡rá ❞✐t❛ q✉❛❞r❛❞❛ s❡ m = n✱ ❡ ♥❡st❡ ❝❛s♦✱ q11, q22, . . . , qnn é ❝❤❛♠❛❞❛ ❞❡ ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧ ❡ qij✱ ❝♦♠ i+j = 1 +n✱ s❡rá ❝❤❛♠❛❞❛ ❞❡ ❞✐❛❣♦♥❛❧

s❡❝✉♥❞ár✐❛ ✭✈❡r ❡♠ ❬✺❪✮✳ ❉✐③❡♠♦s q✉❡ ❛ ♠❛tr✐③ q✉❛❞r❛❞❛ Q❛❜❛✐①♦ é ❞❡ ♦r❞❡♠ n✱ ♦✉ s❡❥❛✱ Qn=     

q11 q12 · · · q1_n q21 q22 · · · q2_n ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ qn1 qn2 · · · qnn

     .

✶✳✸✳✷ ▼❛tr✐③ tr✐❛♥❣✉❧❛r

❯♠❛ ♠❛tr✐③ Tn= [tij]ns❡rá ❞✐t❛ tr✐❛♥❣✉❧❛r s❡ ❢♦r q✉❛❞r❛❞❛ ❡✱ tij = 0✱ ♣❛r❛ i < j ♦✉ tij = 0✱ ♣❛r❛ i > j✳ P♦r ❡①❡♠♣❧♦✱

T3 =





1 0 0

−4 2 0 0 1 5



❡ T4 =



  

3 ₋4 ₋2 1 0 ₋3 5 ₋1 0 0 2 3 0 0 0 ₋1



  

,

sã♦ ♠❛tr✐③❡s tr✐❛♥❣✉❧❛r ✐♥❢❡r✐♦r ❡ s✉♣❡r✐♦r✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

▼❛tr✐③❡s ❈❛♣ít✉❧♦ ✶

✶✳✸✳✸ ▼❛tr✐③ ❞✐❛❣♦♥❛❧

❈❤❛♠❛♠♦s ❞❡ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ t♦❞❛ ♠❛tr✐③ q✉❛❞r❛❞❛ D= [dij]✱ t❛❧ q✉❡ dij = 0✱

s❡ i₆=j✱ ♦✉ s❡❥❛✿

Dn =



   

a11 0 · · · 0

0 a22 _{· · ·} 0

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

0 0 _{· · ·} ann



   

.

✶✳✸✳✹ ▼❛tr✐③ ✐❞❡♥t✐❞❛❞❡

❉✐③❡♠♦s q✉❡ ✉♠❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ ■ ❞❡ ♦r❞❡♠ n é ✉♠❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ q✉❛♥❞♦ aij = 1✱ s❡ i=j✱ ❡ s❡rá ❞❡♥♦t❛❞❛ ♣♦r

In=



   

1 0 _{· · ·} 0 0 1 _{· · ·} 0

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

0 0 _{· · ·} 1



   

.

✶✳✸✳✺ ▼❛tr✐③ ♥✉❧❛

❈❤❛♠❛♠♦s ❞❡ ♠❛tr✐③ ♥✉❧❛ ❛ ♠❛tr✐③ q✉❡ ♣♦ss✉✐ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ✐❣✉❛✐s ❛ ③❡r♦ ❡ s❡rá r❡♣r❡s❡♥t❛❞❛ ♣♦r Om×n✳

Om×n =



   

0 0 _{· · ·} 0 0 0 _{· · ·} 0

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

0 0 _{· · ·} 0



   

.

✶✳✸✳✻ ▼❛tr✐③ tr❛♥s♣♦st❛

❙❡❥❛ A = [aij]m×_n ✉♠❛ ♠❛tr✐③✱ ❝❤❛♠❛♠♦s ❞❡ ♠❛tr✐③ tr❛♥s♣♦st❛ ❞❡ A ❛ ♠❛tr✐③

❝✉❥❛s ❧✐♥❤❛s ♣❛ss❛♠ ❛ s❡r ❛s ❝♦❧✉♥❛s✱ ♦✉ s❡❥❛✱

At_{= [aji]} n×m,

❝♦♠ 1_≤i_≤m ❡ 1_≤j _≤n✳ P♦r ❡①❡♠♣❧♦✱ ❛ ♠❛tr✐③

5 ₋3 0

−2 1 4

♣♦ss✉✐ ❝♦♠♦ tr❛♥s♣♦st❛ ❛ ♠❛tr✐③

▼❛tr✐③❡s ❈❛♣ít✉❧♦ ✶





5 ₋2

−3 1

0 4



.

✶✳✸✳✼ ▼❛tr✐③ s✐♠étr✐❝❛

❈❤❛♠❛♠♦s ❞❡ ♠❛tr✐③ s✐♠étr✐❝❛ ❛ t♦❞❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A ♦♥❞❡At_{=A✳ ◆❛ ✈❡r✲} ❞❛❞❡✱ ♦ ❡❧❡♠❡♥t♦s ❞❡st❛ ♠❛tr✐③ ✜❝❛♠ ❞✐s♣♦st♦s s✐♠❡tr✐❝❛♠❡♥t❡ ❡♠ r❡❧❛çã♦ à ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧✳ P♦r ❡①❡♠♣❧♦✱ ❛ ♠❛tr✐③





3 0 4 0 5 ₋1 4 ₋1 2





é ✉♠❛ ♠❛tr✐③ s✐♠étr✐❝❛✳

✶✳✹ ■❣✉❛❧❞❛❞❡ ❞❡ ♠❛tr✐③❡s

❉✉❛s ♠❛tr✐③ A = [aij]m×n ❡ B = [bij]m×n sã♦ ✐❣✉❛✐s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❛s sã♦ ❞♦ ♠❡s♠♦ t✐♣♦ ❡ ♣♦ss✉❡♠ ❡❧❡♠❡♥t♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ✐❣✉❛✐s✱ ♦✉ s❡❥❛✱ aij =bij✱ ♣❛r❛ t♦❞♦ i, j ♥❛t✉r❛❧✳ P♦r ❡①❡♠♣❧♦✱





a 1 b ₋2 0 ₋4 2 c

2 7 0 1



= 



3 1 2 ₋2 0 ₋4 2 ₋5 2 7 0 1



.

❆ss✐♠✱ a = 3✱ b= 2 ❡ c=₋5✳

✶✳✺ ❖♣❡r❛çõ❡s ❝♦♠ ♠❛tr✐③❡s

❱❡❥❛♠♦s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❛s ♦♣❡r❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s ❞❡ ♠❛tr✐③❡s✿

✶✳✺✳✶ ❆❞✐çã♦ ❞❡ ♠❛tr✐③❡s

❈♦♥s✐❞❡r❡ ✉♠ ❧❛❜♦r❛tór✐♦ ❢❛r♠❛❝ê✉t✐❝♦ q✉❡ ♣r♦❞✉③ ✉♠ ❝❡rt♦ ♠❡❞✐❝❛♠❡♥t♦✳ ❖s ❝✉st♦s r❡❧❛t✐✈♦s à ❝♦♠♣r❛ ❡ tr❛♥s♣♦rt❡ ❞❡ q✉❛♥t✐❞❛❞❡s ❡s♣❡❝í✜❝❛s ❞❛ s✉❜stâ♥❝✐❛ ♥❡❝❡ssár✐❛s ♣❛r❛ ❛ s✉❛ ❡❧❛❜♦r❛çã♦✱ ❛❞q✉✐r✐❞❛s ❡♠ ❞♦✐s ❢♦r♥❡❝❡❞♦r❡s ❞✐st✐♥t♦s sã♦ ❞❛❞♦s ✭❡♠ r❡❛✐s✮ ♣❡❧❛s t❛❜❡❧❛s ❛❜❛✐①♦ ✭✈❡r ❡♠ ❬✺❪✮✳

▼❛tr✐③❡s ❈❛♣ít✉❧♦ ✶

❋♦r♥❡❝❡❞♦r ✶ ♣r❡ç♦ ❞❛ ❝♦♠♣r❛ ❝✉st♦ ❞♦ tr❛♥s♣♦rt❡

❙✉❜stâ♥❝✐❛ ❆ ✸ ✶✺

❙✉❜stâ♥❝✐❛ ❇ ✶✷ ✽

❙✉❜stâ♥❝✐❛ ❈ ✺ ✷

❋♦r♥❡❝❡❞♦r ✷ ♣r❡ç♦ ❞❛ ❝♦♠♣r❛ ❝✉st♦ ❞♦ tr❛♥s♣♦rt❡

❙✉❜stâ♥❝✐❛ ❆ ✻ ✽

❙✉❜stâ♥❝✐❛ ❇ ✾ ✾

❙✉❜stâ♥❝✐❛ ❈ ✸ ✺

❆ ♠❛tr✐③ q✉❡ r❡♣r❡s❡♥t❛ ♦s ❝✉st♦s t♦t❛✐s ❞❡ ❝♦♠♣r❛ ❡ ❞❡ tr❛♥s♣♦rt❡ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s s✉❜stâ♥❝✐❛s A✱ B ❡ C é ❞❛❞❛ ♣♦r✿





3 15 12 8

5 2



+ 



6 8 9 9 3 5



= 



3 + 6 15 + 8 12 + 9 8 + 9

5 + 3 2 + 5



= 



9 23 21 17 8 7



.

❉✐③❡♠♦s q✉❡ ❛ s♦♠❛ ❞❡ ❞✉❛s ♠❛tr✐③❡s ❞❡ ♠❡s♠❛ ♦r❞❡♠✱ Am×_n = [aij] ❡

Bm×n= [bij]✱ é ✉♠❛ ♠❛tr✐③ m×n✱ q✉❡ ❝❤❛♠❛r❡♠♦s ❞❡ A+B✱ ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ ❛s s♦♠❛s ❞♦s ❡❧❡♠❡♥t♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❞❡ A ❡B✳ ■st♦ é✱

A+B = [aij +bij]m×n.

Pr♦♣r✐❡❞❛❞❡s ❞❛ ❛❞✐çã♦ ❞❡ ♠❛tr✐③❡s

❉❛❞❛s ❛s ♠❛tr✐③❡sAm×n✱Bm×n❡Cm×n✱ sã♦ s❛t✐s❢❡✐t❛s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✐✮ ❈♦♠✉t❛t✐✈❛✿ Am×n+Bm×n =Bm×n+Am×n✳

✐✐✮ ❆ss♦❝✐❛t✐✈❛✿ (Am×n+Bm×n) +Cm×n =Am×n+ (Bm×n+Cm×n)✳

✐✐✐✮ ❊❧❡♠❡♥t♦ ♥❡✉tr♦✿ ❊①✐st❡ ✉♠❛ ♠❛tr✐③ 0m×n✱ t❛❧ q✉❡Am×_n+ 0_m×_n=Am×n✳

✐✈✮ ❊❧❡♠❡♥t♦ s✐♠étr✐❝♦✿ P❛r❛ t♦❞❛ ♠❛tr✐③ Am×n✱ ❡①✐st❡ ❛ ♠❛tr✐③ −Am×n✱ t❛❧ q✉❡

Am×n+ (−Am×n) =Om×n✱ ♦♥❞❡ −Am×n = [−aij]m×n ✭✈❡r s❡çã♦ ✶✳✹✳✷✮✳

▼❛tr✐③❡s ❈❛♣ít✉❧♦ ✶

✶✳✺✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠ ❡s❝❛❧❛r ♣♦r ✉♠❛ ♠❛tr✐③

✶✳ ❖ q✉❛❞r♦ ❞❛❞♦ ❛❜❛✐①♦ ♠♦str❛ ❛ ♣r♦❞✉çã♦ ❞❡ tr✐❣♦✱ ❝❡✈❛❞❛✱ ♠✐❧❤♦ ❡ ❛rr♦③ ❡♠ três r❡❣✐õ❡s✱ ❡♠ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ é♣♦❝❛ ❞♦ ❛♥♦ ✭✈❡r ❡♠ ❬✺❪✮

tr✐❣♦ ❝❡✈❛❞❛ ♠✐❧❤♦ ❛rr♦③

❘❡❣✐ã♦ ■ ✶✷✵✵ ✽✵✵ ✺✵✵ ✼✵✵

❘❡❣✐ã♦ ■■ ✻✵✵ ✸✵✵ ✼✵✵ ✾✵✵

❘❡❣✐ã♦ ■■■ ✶✵✵✵ ✶✶✵✵ ✷✵✵ ✹✺✵

❈♦♠ ♦s ✐♥❝❡♥t✐✈♦s ♦❢❡r❡❝✐❞♦s✱ ❡st✐♠❛✲s❡ q✉❡ ❛ s❛❢r❛ ♥♦ ♠❡s♠♦ ♣❡rí♦❞♦ ❞♦ ♣ró✲ ①✐♠♦ ❛♥♦ s❡❥❛ ❞✉♣❧✐❝❛❞❛✳ ❆ ♠❛tr✐③ q✉❡ r❡♣r❡s❡♥t❛ ❛ ❡st✐♠❛t✐✈❛ ❞❡ ♣r♦❞✉çã♦ ♣❛r❛ ♦ ♣ró①✐♠♦ ❛♥♦ é✿

2_·





1200 800 500 700 600 300 700 900 1000 1100 200 450



= 



2400 1600 100 1400 1200 600 1400 1800 2000 2200 400 900



.

✷✳ ❉❛❞❛ ❛ ♠❛tr✐③

A=





3 0

−1 8

5 ₋2



,

❡♥tã♦ ♦ ♣r♦❞✉t♦ ❞❛ ♠❛tr✐③ A ♣❡❧♦ ❡s❝❛❧❛r ✭✲✶✮ s❡rá ❞❛❞♦ ♣♦r✿

(₋1)_·A=





−3 ₋0 1 ₋8

−5 2



.

❆ss✐♠✱ (₋1)_·A=₋A ❡ ❞✐③❡♠♦s q✉❡₋A s❡rá ❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ ♦♣♦st❛ ❞❡ A✳

❙❡❥❛ A = [aij]m×n ✉♠❛ ♠❛tr✐③ ❡ k ✉♠ ♥ú♠❡r♦ r❡❛❧✱ ❡♥tã♦ ❞❡✜♥✐♠♦s ❛ ♠✉❧t✐♣❧✐✲ ❝❛çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦r ✉♠❛ ♠❛tr✐③ ♣❡❧❛ ♥♦✈❛ ♠❛tr✐③ ❛❜❛✐①♦

k_·A = [k_·aij]m×n.

Pr♦♣r✐❡❞❛❞❡s ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠ ❡s❝❛❧❛r ♣♦r ✉♠❛ ♠❛tr✐③

❉❛❞❛s ❛s ♠❛tr✐③❡s Am×n✱ Bm×n ❡ Cm×n ❡ s❡❥❛♠ α ❡β ♥ú♠❡r♦s r❡❛✐s✱ ❡♥tã♦ sã♦ ✈❡r❞❛❞❡✐r❛s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ✭✈❡r ❡♠ ❬✶✵❪✮✿

✐✮ α(βAm×n) = (αβ)Am×n✳

▼❛tr✐③❡s ❈❛♣ít✉❧♦ ✶

✐✐✮ (α+β)Am×n=αAm×n+βAm×n✳

✐✐✐✮ α(Am×_n+Bm×_n) =αAm×_n+αBm×n✳

✐✈✮ 1.Am×n=Am×n✳

✶✳✺✳✸ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s

✶✳ ❯♠ ❡♠♣r❡sár✐♦ ❢♦r♥❡❝❡ ♠❡♥s❛❧♠❡♥t❡ ❛❧✐♠❡♥t♦s ❛ ❞♦✐s ♦r❢❛♥❛t♦s✳ P❛r❛ ♦ ♣r✐♠❡✐r♦ ♦r❢❛♥❛t♦ sã♦ ❞♦❛❞♦s ✷✺ ❦❣ ❞❡ ❛rr♦③✱ ✷✵ ❦❣ ❞❡ ❢❡✐❥ã♦✱ ✸✵ ❦❣ ❞❡ ❝❛r♥❡ ❡ ✸✷ ❦❣ ❞❡ ❜❛t❛t❛✳ P❛r❛ ♦ s❡❣✉♥❞♦✱ ✷✽ ❦❣ ❞❡ ❛rr♦③✱ ✷✹ ❦❣ ❞❡ ❢❡✐❥ã♦✱ ✸✺ ❦❣ ❞❡ ❝❛r♥❡ ❡ ✸✽ ❦❣ ❞❡ ❜❛t❛t❛✳ ❖ ❡♠♣r❡sár✐♦ ❢❛③ ❛ ❝♦t❛çã♦ ❞❡ ♣r❡ç♦s ❡♠ ❞♦✐s s✉♣❡r♠❡r❝❛❞♦s ❡ r❡♣r❡s❡♥t❛✲♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ t❛❜❡❧❛ ❛❜❛✐①♦ ✭✈❡r ❡♠ ❬✻❪✮✿

P❘❖❉❯❚❖ ✭✶ ❦❣✮ ❙❯P❊❘▼❊❘❈❆❉❖ ✶ ❙❯P❊❘▼❊❘❈❆❉❖ ✷

❆rr♦③ ✶✱✵✵ ✶✱✵✵

❋❡✐❥ã♦ ✶✱✺✵ ✶✱✷✵

❈❛r♥❡ ✻✱✵✵ ✼✱✵✵

❇❛t❛t❛ ✵✱✽✵ ✵✱✻✵

❉❡t❡r♠✐♥❡ ♦ ❣❛st♦ ♠❡♥s❛❧ ❞❡ss❡ ❡♠♣r❡sár✐♦✱ ♣♦r ♦r❢❛♥❛t♦✱ s✉♣♦♥❞♦ q✉❡ t♦❞♦s ♦s ♣r♦❞✉t♦s s❡❥❛♠ ❛❞q✉✐r✐❞♦s ♥♦ ♠❡s♠♦ ❡st❛❜❡❧❡❝✐♠❡♥t♦ ❡ q✉❡ ❡st❡ r❡♣r❡s❡♥t❡ ❛ ♠❡❧❤♦r ♦♣çã♦ ❞❡ ❝♦♠♣r❛✳

❈❤❛♠❛♥❞♦ ❞❡ A ❛ ♠❛tr✐③ q✉❡ r❡♣r❡s❡♥t❛ ❛ ❝♦♠♣r❛ ❞♦s ♣r♦❞✉t♦s ♣❛r❛ ♦s ❞♦✐s ♦r❢❛♥❛t♦s✿

A=

25 20 30 32 28 24 35 38

.

❆❣♦r❛✱ s❡❥❛ B ❛ ♠❛tr✐③ q✉❡ r❡♣r❡s❡♥t❛ ♦ ♣r❡ç♦ ❞♦s ♣r♦❞✉t♦s ♥♦s ❞♦✐s s✉♣❡r♠❡r✲ ❝❛❞♦s✿

B =



  

1,00 1,00 1,50 1,20 6,00 7,00 0,80 0,60



  

.

❆❣♦r❛✱ ✈❛♠♦s ❝❛❧❝✉❧❛r ♦ ❣❛st♦ ♠❡♥s❛❧ ❞♦ ❡♠♣r❡sár✐♦✳ P❛r❛ ♦ ♣r✐♠❡✐r♦ ♦r❢❛♥❛t♦✱ t❡♠♦s✿

• s✉♣❡r♠❡r❝❛❞♦ ✶✿ 25.1,00 + 20.1,00 + 30.6,00 + 32.0,80 = 260,60✳

• s✉♣❡r♠❡r❝❛❞♦ ✷✿ 25.1,00 + 20.1,20 + 30.7,00 + 32.0,60 = 278,20✳

P❛r❛ ♦ s❡❣✉♥❞♦ ♦r❢❛♥❛t♦✱ t❡♠♦s✿

▼❛tr✐③❡s ❈❛♣ít✉❧♦ ✶

• s✉♣❡r♠❡r❝❛❞♦ ✶✿ 28.1,00 + 24.1,00 + 35.6,00 + 38.0,80 = 304,40✳

• s✉♣❡r♠❡r❝❛❞♦ ✷✿ 28.1,00 + 24.1,20 + 35.7,00 + 38.0,60 = 324,60✳

❘❡♣r❡s❡♥t❛r❡♠♦s ♣❡❧❛ ♠❛tr✐③ C ♦s ✈❛❧♦r❡s ♦❜t✐❞♦s ❛❝✐♠❛✳ ❆ss✐♠✱

C =

260,60 278,20 304,40 324,60

.

P♦rt❛♥t♦✱ ❛ ♠❡❧❤♦r ♦♣çã♦ é ❝♦♠♣r❛r ♥♦ ♣r✐♠❡✐r♦ s✉♣❡r♠❡r❝❛❞♦✳

◆♦t❡ q✉❡ ✉t✐❧✐③❛♥❞♦ ❛♣❡♥❛s ❛s ♠❛tr✐③❡s✱ t❡♠♦s✿

25 20 30 32 28 24 35 38

2×4 .



  

1,00 1,00 1,50 1,20 6,00 7,00 0,80 0,60



  

4×2

=

260,60 278,20 304,40 324,60

2×2 .

❉❡ ✉♠ ♠♦❞♦ ❣❡r❛❧✱ ❞❛❞❛s ❛s ♠❛tr✐③❡s A = [aij]m×n ❡ B = [bij]n×p✱ ❞❡✜♥✐♠♦s

❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛s ♠❛tr✐③❡s A ♣♦r B ♣❡❧❛ ♠❛tr✐③ Am×n·Bn×p = Cm×p✱ t❛❧ q✉❡

C = [cij]m×p✱ ♦♥❞❡

cij =

n

X

k=1

aik_·bkj =ai1·b1_j+ai2·b2_j +ai3·b3_j+· · ·+aip·bpj,

❝♦♠ 1_≤i_≤m ❡ 1_≤j _≤n✳

◆♦t❡ q✉❡ ♥ã♦ ❝♦♥s❡❣✉✐rí❛♠♦s ❢❛③❡r ♦ ♣r♦❞✉t♦ ❞❛ ♠❛tr✐③❡s ❞❛❞❛s ❛❜❛✐①♦✱



  

2 ₋1

−3 4

0 ₋2 1 0



  

4×2

·





1 ₋2 3 1 0 ₋4





3×2

♣♦rq✉❡ ♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❞❛ ♣r✐♠❡✐r❛ ♠❛tr✐③ é ❞✐❢❡r❡♥t❡ ❞♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s ❞❛ s❡❣✉♥❞❛ ♠❛tr✐③✳

Pr♦♣r✐❡❞❛❞❡s ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s ✭✈❡r ❡♠ ❬✶✵❪✮

✐✮ ❆ss♦❝✐❛t✐✈❛✿ ❙❡❥❛♠ ❛s ♠❛tr✐③❡s A= [aij]m×n✱B = [bij]_n×_p ❡ C = [cij]_p×q✱ t❡♠♦s✿

(Am×n·Bn×p)·Cp×q =Am×n·(Bn×p·Cp×q).

▼❛tr✐③❡s ❈❛♣ít✉❧♦ ✶

✐✐✮ ❉✐str✐❜✉t✐✈❛ à ❞✐r❡✐t❛✿

❙❡❥❛♠ ❛s ♠❛tr✐③❡s A= [aij]m×n✱ B = [bij]m×n ❡ C = [cij]n×p✱ t❡♠♦s✿

(Am×_n+Bm×_n)·Cn×_p =Am×_n·Cn×_p+Bm×_n·Cn×p.

✐✐✐✮ ❉✐str✐❜✉t✐✈❛ à ❡sq✉❡r❞❛✿

❙❡❥❛♠ ❛s ♠❛tr✐③❡s A= [aij]m×n✱ B = [bij]m×n ❡ C = [cij]p×m✱ t❡♠♦s✿

Cp×_m·(Am×_n+Bm×_n) = Cp×_m·Am×_n+Cp×_m·Bm×n.

✶✳✺✳✹ ▼❛tr✐③❡s ✐♥✈❡rtí✈❡✐s ✭♦✉ ✐♥✈❡rsí✈❡✐s✮

❉✐③❡♠♦s q✉❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A ❞❡ ♦r❞❡♠ n é ✐♥✈❡rsí✈❡❧✱ s❡ ❡①✐st✐r ✉♠❛ ♠❛tr✐③ B✱ ❞❡ ♠❡s♠❛ ♦r❞❡♠✱ t❛❧ q✉❡✿

An.Bn =Bn.An =In. ❘❡♣r❡s❡♥t❛r❡♠♦s ❛ ♠❛tr✐③ ✐♥✈❡rs❛ B ♣♦r A−1✱ ❛ss✐♠✿

An.A−1

n =A

−1

n .An=In.

❙❡ ❛ ♠❛tr✐③ A♥ã♦ ❢♦r ✐♥✈❡rsí✈❡❧✱ ❞✐③❡♠♦s q✉❡ ❡❧❛ é ✉♠❛ ♠❛tr✐③ s✐♥❣✉❧❛r ✭✈❡r ❡♠ ❬✼❪✮✳

❚❡♦r❡♠❛✿ ❙❡ A é ✐♥✈❡rsí✈❡❧✱ ❡♥tã♦ B é ❛ ú♥✐❝❛ ♠❛tr✐③ t❛❧ q✉❡

An.Bn =Bn.An =In.

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛ ✉♠❛ ♦✉tr❛ ♠❛tr✐③ C t❛❧ q✉❡

An.Cn =Cn.An =In.

❊♥tã♦✱ t❡♠♦s q✉❡✿

Cn=Cn.In =Cn.(An.Bn) = (Cn.An).Bn =In.Bn=Bn

▼❛tr✐③❡s ❈❛♣ít✉❧♦ ✶

P♦r ❡①❡♠♣❧♦✱ ❛ ♠❛tr✐③

3 1 5 2

é ✐♥✈❡rtí✈❡❧ ❡ s✉❛ ✐♥✈❡rs❛ é

2 ₋1

−5 3

.

❉❡ ❢❛t♦✱ ❢❛③❡♥❞♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛s ♠❛tr✐③❡s ❞❛❞❛s ♦❜t❡♠♦s✿

3 1 5 2 .

2 ₋1

−5 3

=

6₋5 ₋3 + 3 10₋10 ₋5 + 6

= 1 0 0 1 .

❏á ❛ ♠❛tr✐③

−2 3

−4 6

♥ã♦ ♣♦ss✉✐ ✐♥✈❡rs❛✳

❉❡ ❢❛t♦✱ s✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛ ❛ ♠❛tr✐③

A−1

= a c b d . ▲♦❣♦✱

An.A−1 n =In

−2 3

−4 6

. a c b d = 1 0 0 1

−2a+ 3b ₋2c+ 3d

−4a+ 6b ₋4c+ 6d

= 1 0 0 1 ❆ss✐♠✱

−2a+ 3b = 1

−4a+ 6b = 0 ❡

−2c+ 3d= 0

−4c+ 6d= 1 .

❆♦ t❡♥t❛r♠♦s r❡s♦❧✈❡r q✉❛❧q✉❡r ✉♠ ❞♦s s✐st❡♠❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ♣r✐♠❡✐r♦ s✐s✲ t❡♠❛✱ ❡ ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ s✉❛ r❡s♦❧✉çã♦✱ ✈❛♠♦s ❞✐✈✐❞✐r ❛ s❡❣✉♥❞❛ ❡q✉❛çã♦ ♣♦r ✷✱ ❧♦❣♦✿

−2a+ 3b= 1

−2a+ 3b= 0 .

❆ss✐♠✱ ♦❜t❡♠♦s q✉❡ 1 = 0✱ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ P♦rt❛♥t♦✱ ♥ã♦ ❡①✐st❡ ❛ ♠❛tr✐③

✐♥✈❡rs❛ ❞❡ A✳

Pr♦♣r✐❡❞❛❞❡ ❞❛s ♠❛tr✐③❡s ✐♥✈❡rtí✈❡✐s✿

❙❡ A❡B sã♦ ♠❛tr✐③❡s ✐♥✈❡rtí✈❡✐s✱ ❡♥tã♦ A_·B é ✐♥✈❡rtí✈❡❧ ❡(A_·B)−1

=B−1

·A−1✳

❈❛♣ít✉❧♦ ✷

❉❊❚❊❘▼■◆❆◆❚❊❙

✷✳✶ ❆s♣❡❝t♦s ❤✐stór✐❝♦s

❖ sé❝✉❧♦ ❳■❳ ❢♦✐ ♠❛r❝❛❞♦ ♣♦r ❣r❛♥❞❡s ❛✈❛♥ç♦s ♥❛ ár❡❛ ❞❛ ♣❡sq✉✐s❛ ♠❛t❡♠át✐❝❛✳ ❊r❛ ♦ á♣✐❝❡ ❞❡ ✉♠ ♣r♦❝❡ss♦ q✉❡ ✈✐♥❤❛ ❛❝♦♥t❡❝❡♥❞♦ ❞❡s❞❡ ❛ é♣♦❝❛ ❞❡ ◆❡✇t♦♥✱ ♥❛ ■♥❣❧❛t❡rr❛✱ ❡ ▲❡✐❜♥✐③✱ ♥❛ ❆❧❡♠❛♥❤❛✱ ❞♦✐s sé❝✉❧♦s ❛♥t❡s✳ ❯♠❛ ❞❛s ❢❡rr❛♠❡♥t❛s ❡st✉✲ ❞❛❞❛s ♥❡ss❛ é♣♦❝❛ ❡r❛ ♦ ✉s♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✳ ❊♠❜♦r❛ ❛ ♥♦çã♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s ❡st✐✈❡ss❡ ♣r❡s❡♥t❡ ❡♥tr❡ ♦s ❝❤✐♥❡s❡s✱ q✉❡ ✉s❛✈❛♠ s❡✉ ❝á❧❝✉❧♦ ♣❛r❛ r❡s♦❧✈❡r s✐st❡♠❛s ❧✐♥❡❛r❡s✳ ❙♦♠❡♥t❡ ♥♦ ✜♥❛❧ ❞♦ ✱ ♦ ♠❛t❡♠át✐❝♦ ❥❛♣♦♥ês ❞♦ sé✲ ❝✉❧♦ ❳❱■■✱ ❙❡❦✐ ❑♦✇❛✱ s✐st❡♠❛t✐③♦✉ t❛❧ ♣r♦❝❡❞✐♠❡♥t♦✳ ❚r❛t❛✈❛✲s❡ ❞❡ ❞✉❛s ❡q✉❛çõ❡s ❛ ❞✉❛s ✐♥❝ó❣♥✐t❛s✱ ❢♦r♠❛♥❞♦ ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s✳ ◆❛ ♥♦t❛çã♦ ❛t✉❛❧✱

t❡♠♦s✿

ax+by=c dx+ey=f , ♦♥❞❡✿

x= c.e−b.f

a.e₋b.d ❡ y=

a.f₋c.d

a.e₋b.d✳ ✭s❡ a.e−b.d6= 0✮

❚❛♠❜é♠ ♥❡ss❡ sé❝✉❧♦✱ ♥♦ ♦❝✐❞❡♥t❡✱ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s ❡st❛✈❛ ❝♦♠❡✲ ç❛♥❞♦ ❞❡ ♠♦❞♦ ♠❛✐s s✐st❡♠át✐❝♦✱ ✉♠❛ ❞é❝❛❞❛ ❞❡♣♦✐s ❞❡ ▲❡✐❜♥✐③ t❡r ❡s❝r✐t♦ ✉♠ tr❛❜❛❧❤♦ s♦❜r❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ❝♦♠ três ❡q✉❛çõ❡s ❡ três ✐♥❝ó❣♥✐t❛s✳ ❊♠ ♠❡❛❞♦s ❞♦ sé❝✉❧♦ ❳❱■■■ ♦ ❡s❝♦❝ês ❈♦❧✐♥ ▼❛❝ ▲❛✉r✐♥ ❡ ♦ s✉✐ç♦ ●❛❜r✐❡❧ ❈r❛♠❡r✱ ✐♥❞❡♣❡♥❞❡♥t❡s ✉♠ ❞♦ ♦✉tr♦✱ ❞❡s❝♦❜r✐r❛♠ ✉♠❛ r❡❣r❛ ♣❛r❛ r❡s♦❧✈❡r s✐st❡♠❛s ❧✐♥❡❛r❡s ❞❡ ♥ ❡q✉❛çõ❡s ❡ ♥ ✐♥❝ó❣♥✐t❛s✱ q✉❡ ✜❝♦✉ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ r❡❣r❛ ❞❡ ❈r❛♠❡r✳

❯♠ ♣♦✉❝♦ ♠❛✐s t❛r❞❡✱ ♠❛s ❛✐♥❞❛ ♥♦ ♠❡s♠♦ sé❝✉❧♦✱ ♦ ❛❧❡♠ã♦ ❈❛r❧ ❋r✐❡❞r✐❝❤ ●❛✉ss ♥♦♠❡♦✉ ❝♦♠♦ ❞❡t❡r♠✐♥❛♥t❡s ❛s ❡①♣r❡ssõ❡s ♥✉♠ér✐❝❛s ❛❞✈✐♥❞❛s ❞♦s s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❡✱ ♠❛✐s t❛r❞❡✱ ♦s ♠❛t❡♠át✐❝♦s ❢r❛♥❝❡s❡s ➱t✐❡♥♥❡ ❇❡③♦✉t ❡ ❆❧❡①❛♥❞r❡ ❱❛♥❞❡r♠♦♥❞❡✱ ❝♦♥str✉ír❛♠ ❛ t❡♦r✐❛ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s s❡♣❛r❛❞❛ ❞♦ ❡st✉❞♦ ❞♦s s✐s✲ t❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s✳ ❊♠ ✶✽✶✷✱ ❈❛✉❝❤② ✉t✐❧✐③♦✉ ♦ t❡r♠♦ ✧❞❡t❡r♠✐♥❛♥t❡✧♥✉♠ tr❛❜❛❧❤♦ ♥♦ q✉❛❧ r❡s✉♠✐✉ ♦ ❛ss✉♥t♦ ❡ ♠❡❧❤♦r♦✉ ❛ ♥♦t❛çã♦ ❡♠♣r❡❣❛❞❛ ❛té ❡♥tã♦✳

❉❡t❡r♠✐♥❛♥t❡s ❈❛♣ít✉❧♦ ✷

❆ t❡♦r✐❛ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s q✉❡ ✉t✐❧✐③❛♠♦s ❤♦❥❡✱ s❡ ❞❡✈❡ ❛♦ ❛❧❡♠ã♦ ❈❛r❧ ●✉s✲ t❛✈ ❏❛❝♦❜ ✭✶✽✵✹✲✶✽✺✶✮✱ q✉❡ ❛❝r❡❞✐t❛✈❛ ♠✉✐t♦ ♥♦ ✉s♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ❝♦♠♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❡✜❝❛③ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♥❛ ❋ís✐❝❛ ❡ ❊❝♦♥♦♠✐❛ ✭✈❡r ❡♠ ❬✹❪✮✳

✷✳✷ ❉❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛

❉❡t❡r♠✐♥❛♥t❡ é ✉♠ ♥ú♠❡r♦ r❡❛❧ ❛ss♦❝✐❛❞♦ ❛ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❞❡ ♦r❞❡♠ ♥✱ ♦♥❞❡ n é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❘❡♣r❡s❡♥t❛♠♦s ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ A ♣♦r detA♦✉ _|A_|✳

◆❡st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛♣❡♥❛s ❛ ❞❡✜♥✐çã♦ ❞♦ ❞❡t❡r♠✐♥❛♥t❡ ♣❛r❛ ❛s ♠❛✲ tr✐③❡s ❝♦♠ n_{∈ {}1,2,3_} ♠❛tr✐③ ✭✈❡r ❡♠ ❬✾❪✮✳

• P❛r❛ n= 1✱ t❡♠♦sA= [a11]⇒detA=a11✳

• P❛r❛ n= 2✱ t❡♠♦sA=

a11 a12 a21 a22

⇒detA=a11·a22−a12·a21✳

• P❛r❛ n= 3✱ t❡♠♦sA=





a11 a12 a13 a21 a22 a23 a31 a32 a33



,❡♥tã♦✿

detA=a11a22a33+a12a23a31+a13a21a32−a13a22a31−a11a23a32−a12a21a33.

❱❡❥❛♠♦s ♦s ❡①❡♠♣❧♦ ❛❜❛✐①♦✱ ♣❛r❛ ✉♠❛ ♠❡❧❤♦r ✜①❛çã♦ ❞♦ ❝á❧❝✉❧♦ ❞♦s ❞❡t❡r♠✐✲ ♥❛♥t❡s✿

✶✳ ❙❡A= [₋√2]✱ ❡♥tã♦ detA=₋√2.

✷✳ ❙❡ B =

2 ₋3 4 ₋7

✱ ❡♥tã♦ detB = 2_·(₋7)₋(₋3)_·4 =₋14 + 12 = ₋2✳

✸✳ ❙❡❥❛ ❛ ♠❛tr✐③ C =





2 5 ₋2

−1 0 4

0 1

2 0



✱ ❡♥tã♦✿

|C_|= 2_·0_·0 + 5_·4_·0 + (₋2)_·(₋1)_·1

2 −(−2)·0·0−2·4· 1

2 −5·(−1).0

|C_|= 0 + 0 + 1₋0₋4₋0

|C_|=₋3.

❉❡t❡r♠✐♥❛♥t❡s ❈❛♣ít✉❧♦ ✷

✷✳✸ ❉❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ tr✐❛♥❣✉❧❛r

➱ ✐♠♣♦rt❛♥t❡ ♦❜s❡r✈❛r q✉❡✱ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ q✉❛❧q✉❡r ♠❛tr✐③ tr✐❛♥❣✉❧❛r s❡rá ❞❛❞♦ ♣❡❧♦ ♣r♦❞✉t♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❛ ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧✱ ♦✉ s❡❥❛✱

detA=a11a22. . . ann.

P♦r ❡①❡♠♣❧♦✱ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ tr✐❛♥❣✉❧❛r s✉♣❡r✐♦r ❞❛❞❛ ❛❜❛✐①♦ é ❞❛❞♦ ♣♦r✿

2 0 0 0 1 ₋3 0 0 7 1 1 0 2 0 1 4

= 2_·(₋3)_·1_·4 = ₋24

✷✳✹ Pr♦♣r✐❡❞❛❞❡s ❞♦s ❞❡t❡r♠✐♥❛♥t❡s

❆♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ✭✈❡r ❡♠ ❬✶✵❪✮✿

✭✶✮ detA= 0✱ q✉❛♥❞♦ ❛ ♠❛tr✐③ A ♣♦ss✉✐r ✉♠❛ ✜❧❛ ✭❧✐♥❤❛ ♦✉ ❝♦❧✉♥❛✮ ♥✉❧❛✳ ✭✷✮ detA= 0✱ q✉❛♥❞♦ ❛ ♠❛tr✐③ A ♣♦ss✉✐r ❞✉❛s ❧✐♥❤❛s ✭♦✉ ❝♦❧✉♥❛s✮ ✐❣✉❛✐s✳ ✭✸✮ det(k_·A) =kn_·detA✳

✭✹✮ det(A_·B) = detA_·detB✳ ✭✺✮ detA=detAt_✳

✭✻✮ ❙❡B é ❛ ♠❛tr✐③ ♦❜t✐❞❛ ❞❡Atr♦❝❛♥❞♦✲s❡ ❛i✲és✐♠❛ ❧✐♥❤❛ ✭♦✉ ❝♦❧✉♥❛✮ ♣❡❧❛j✲és✐♠❛ ❧✐♥❤❛ ✭❝♦❧✉♥❛✮✱ ❡♥tã♦ detB =₋detA✳

✭✼✮ ❯♠❛ ♠❛tr✐③ A é ✐♥✈❡rtí✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ detA₆= 0✳

✭✽✮ ❙❡ A é ✉♠❛ ♠❛tr✐③ ✐♥✈❡rtí✈❡❧ ❡♥tã♦✱ detA−1

= 1

detA✳

Pr♦✈❛✿ ❙❡❥❛ A ✉♠❛ ♠❛tr✐③ ✐♥✈❡rtí✈❡❧ ❝✉❥❛ ✐♥✈❡rs❛ s❡❥❛ ❞❛❞❛ ♣♦r A−1✳ ❚❡♠♦s q✉❡✿

A_·A−1

=In

det(A_·A−1

) =det(In)

P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❇✐♥❡t✱ t❡♠♦s✿

detA_·detA−1

=detIn

❉❡t❡r♠✐♥❛♥t❡s ❈❛♣ít✉❧♦ ✷

❆ss✐♠✱

detA_·detA−1

= 1

detA−1

= 1

detA s❡ detA₆= 0✳

❖❜s❡r✈❛çõ❡s✿

✶✳ ❙❡❥❛

A=

a c b d

✉♠❛ ♠❛tr✐③ ✐♥✈❡rtí✈❡❧✱ ❡♥tã♦

A−1

= 1

detA

d ₋b

−c a

.

✷✳ ❱❛♠♦s r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ A_·X = B✱ s❛❜❡♥❞♦ q✉❡ ❛ ♠❛tr✐③ A é ✐♥✈❡rtí✈❡❧✱ ♦✉ s❡❥❛✱ detA₆= 0✳

❙❡❥❛ A−1

❛ ♠❛tr✐③ ✐♥✈❡rs❛ ❞❡ A✱ ❡♥tã♦✿

A−1

·(A_·X) =A−1

·B

(A−1

·A)_·X =A−1

·B In_·X =A−1

·B X =A−1

·B.

❈❛♣ít✉❧♦ ✸

❙■❙❚❊▼❆❙ ▲■◆❊❆❘❊❙

✸✳✶ ❆s♣❡❝t♦s ❤✐stór✐❝♦s

❆s ❛♣❛r✐çõ❡s ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s sã♦ ♣♦✉❝❛s ❡s♣❡❝✐❛❧♠❡♥t❡ ♥❛ ♠❛t❡✲ ♠át✐❝❛ ♦❝✐❞❡♥t❛❧ ❛♥t✐❣❛✳ ❈♦♥t✉❞♦✱ ♥♦ ♦r✐❡♥t❡✱ ♦s ❝❤✐♥❡s❡s ❝♦♠ s❡✉ ❣♦st♦ ❡s♣❡❝✐❛❧ ♣♦r ❞✐❛❣r❛♠❛s✱ r❡♣r❡s❡♥t❛r❛♠ ♦s s✐st❡♠❛s ❧✐♥❡❛r❡s ♣♦r ♠❡✐♦ ❞❡ s❡✉s ❝♦❡✜❝✐❡♥t❡s ❡s❝r✐t♦s ❝♦♠ ❜❛rr❛s ❞❡ ❜❛♠❜✉ s♦❜r❡ ♦s q✉❛❞r❛❞♦s ❞❡ ✉♠ t❛❜✉❧❡✐r♦ ❡ ❝♦♠ ✐ss♦✱ ❞❡s❝♦❜r✐r❛♠ ♦ ♠ét♦❞♦ ❞❡ r❡s♦❧✉çã♦ ♣♦r ❡❧✐♠✐♥❛çã♦ ✭❝♦♥s✐st❡ ❡♠ t♦r♥❛r ♥✉❧♦ ❝♦❡✜❝✐❡♥t❡s ♣♦r ♠❡✐♦ ❞❡ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s✮✳ ❊①❡♠♣❧♦s ❞❡ t❛❧ ♣r♦❝❡❞✐♠❡♥t♦ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ♥♦ ❧✐✈r♦ ❝❤✐♥ês ◆♦✈❡ ❝❛♣ít✉❧♦s s♦❜r❡ ❛ ❛rt❡ ❞❛ ♠❛t❡♠át✐❝❛✱ ❞❡ ❈❤✉✐✲❈❤❛♥❣ ❙✉❛♥✲❙❤✉✱ ✉♠❛ ♦❜r❛ q✉❡ ❞❛t❛ ♣r♦✈❛✈❡❧♠❡♥t❡ ❞♦ sé❝✉❧♦ ■■■ ❛✳❈✳✭✈❡r ❡♠ ❬✼❪✮✳

❆♣❡♥❛s ❡♠ ✶✻✽✸✱ ♥✉♠ tr❛❜❛❧❤♦ ❞♦ ❥❛♣♦♥ês ❙❡❦✐ ❑♦✇❛✱ é q✉❡ ❛ ✐❞❡✐❛ ❞❡ ❞❡t❡r✲ ♠✐♥❛♥t❡ ✭❝♦♠♦ ♣♦❧✐♥ô♠✐♦ ❛ss♦❝✐❛❞♦ ❛ ✉♠ q✉❛❞r❛❞♦ ❞❡ ♥ú♠❡r♦s✮ ✈❡✐♦ à t♦♥❛✳ ❚❛❧ ❥❛♣♦♥ês ❝❤❡❣♦✉ ❛ ❡ss❛ ♥♦çã♦ ♣♦r ♠❡✐♦ ❞♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s✱ s✐st❡♠❛t✐③❛♥❞♦ ♦ ❛♥t✐❣♦ ♣r♦❝❡❞✐♠❡♥t♦ ❝❤✐♥ês✱ ♣❛r❛ ❝❛s♦s ❞❡ ❞✉❛s ❡q✉❛çõ❡s ❛♣❡♥❛s ✭✈❡r ❡♠ ❬✼❪✮✳

✸✳✷ ❉❡✜♥✐çã♦

❯♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❝♦♠ ♠ ❡q✉❛çõ❡s ❡ ♥ ✐♥❝ó❣♥✐t❛s é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡q✉❛çõ❡s ❞❛ ❢♦r♠❛✿

    

   

a11x1 + a12x2 + · · · + a1nxn = b1 a21x1 + a22x2 + · · · + a2nxn = b2

✳✳✳ ✳✳✳ ✳✳✳ + ✳✳✳ + ✳✳✳ = ✳✳✳

am1x1 + am2x2 + · · · + amnxn = bm

✭✸✳✶✮

♦♥❞❡ aij ❡ bi sã♦ ♥ú♠❡r♦s r❡❛✐s✱ ❝♦♠ i= 1, . . . , m❡ j = 1, . . . , n✳

❯♠❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ✭✸✳✶✮ é ✉♠❛ ♥✲✉♣❧❛

❙✐st❡♠❛s ▲✐♥❡❛r❡s ❈❛♣ít✉❧♦ ✸

X = (x1, x2, . . . , xn) ♦✉ X = [x1, x2, . . . , xn]

q✉❡ s❛t✐s❢❛③ ❝❛❞❛ ✉♠❛ ❞❛s ♠ ❡q✉❛çõ❡s ✭✈❡r ❡♠ ❬✶❪✮✳

✸✳✸ ❋♦r♠❛ ♠❛tr✐❝✐❛❧

P♦❞❡♠♦s ❛ss♦❝✐❛r ♠❛tr✐③❡s ❛ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r✳ ◆♦ s✐st❡♠❛ ✸✳✶ t❡♠♦s ❛ ♠❛tr✐③

M =     

a11 a12 · · · a1_n b1 a21 a22 · · · a2_n b2 ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ am1 am2 · · · amn bm

     .

❖❜s❡r✈❡ q✉❡ ❝❛❞❛ ❧✐♥❤❛ é ❢♦r♠❛❞❛✱ ♦r❞❡♥❛❞❛♠❡♥t❡✱ ♣❡❧♦s ❝♦❡✜❝✐❡♥t❡s ❡ ♣❡❧♦s t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ ❝❛❞❛ ❡q✉❛çã♦✳ ❊ss❛ ♠❛tr✐③ M é ❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ ❝♦♠♣❧❡t❛✳ ❚❡♠♦s t❛♠❜é♠ ❛s ♠❛tr✐③❡s✿

A=     

a11 a12 · · · a1_n a21 a22 · · · a2_n ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ am1 am2 · · · amn

     ,

❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ ❞♦s ❝♦❡✜❝✐❡♥t❡s❀

X=      x1 x2 ✳✳✳ xn     

❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ ❞❛s ✐♥❝ó❣♥✐t❛s❀

B =      b1 b2 ✳✳✳ bm     

❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ ❞♦s t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s✳

P♦rt❛♥t♦✱ s❡ ♠✉❧t✐♣❧✐❝❛r♠♦s ❛s ♠❛tr✐③❡s A ♣♦r X ♦❜t❡♠♦s ❛ ♠❛tr✐③ B✳ ❖✉ s❡❥❛✱



   

a11 a12 · · · a1_n a21 a22 _{· · ·} a2n ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ am1 am2 · · · amn

❙✐st❡♠❛s ▲✐♥❡❛r❡s ❈❛♣ít✉❧♦ ✸

❊ ❝❤❛♠❛♠♦s ❞❡ ❢♦r♠❛ ♠❛tr✐❝✐❛❧ ❛ ❡ss❛ ♠❛♥❡✐r❛ ❞❡ ❡s❝r❡✈❡r ♦ s✐st❡♠❛ ❧✐♥❡❛r✳ P♦r ❡①❡♠♣❧♦✱

x+y= 5

4x₋3y=₋4 ⇔

1 1 4 ₋3

.

x y

=

5

−4

.

✸✳✹ ❙✐st❡♠❛ ▲✐♥❡❛r ❍♦♠♦❣ê♥❡♦

❙❡ b1 =b2 =. . . =bm = 0✱ ❞✐③❡♠♦s q✉❡ ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s é ❤♦♠♦✲ ❣ê♥❡♦✳ ◆♦t❡ q✉❡ ❛ ♥✲✉♣❧❛ (0,0,_{· · ·} ,0)é s❡♠♣r❡ ✉♠❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❤♦♠♦❣ê♥❡♦✱

❡ ❡st❛ n✲✉♣❧❛ s❡rá ❝❤❛♠❛❞❛ ❞❡ s♦❧✉çã♦ tr✐✈✐❛❧✳ ❉❡ss❡ ♠♦❞♦✱ t❡♠♦s q✉❡ ♦ s✐st❡♠❛

    

   

a11x1 + a12x2 + · · · + a1nxn = 0 a21x1 + a22x2 + · · · + a2nxn = 0

✳✳✳ ✳✳✳ ✳✳✳ + ✳✳✳ + ✳✳✳ = ✳✳✳

am1x1 + am2x2 + · · · + amnxn = 0

é ❧✐♥❡❛r ❡ ❤♦♠♦❣ê♥❡♦✳ P♦r ❡①❡♠♣❧♦✱

3x₋2y= 0 2x+ 4y= 0 ❡

 



x₋2y₋5z₋t = 0 2x+ 4y₋2z+ 0t = 0

−3x+ 2y+ 0z3t= 0 .

✸✳✺ ❙✐st❡♠❛ ▲✐♥❡❛r ❊q✉✐✈❛❧❡♥t❡

❉✐③❡♠♦s q✉❡ ❞♦✐s s✐st❡♠❛s ❧✐♥❡❛r❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s q✉❛♥❞♦ ❡❧❡s ❛♣r❡s❡♥t❛♠ ❛ ♠❡s♠❛ s♦❧✉çã♦✳ P♦r ❡①❡♠♣❧♦✱ ♦s s✐st❡♠❛s✿

x₋2y=₋3 2x+ y = 4 ❡

3x₋4y =₋5

x+ 2y= 5

sã♦ ❡q✉✐✈❛❧❡♥t❡s✱ ♣♦✐s ❛♠❜♦s ❛♣r❡s❡♥t❛♠ ♦ ♠❡s♠♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ S = _{(1,2)_}

✭❛♣ós ❛ s❡çã♦ ✸✳✼ t❡r❡♠♦s ❝♦♥t❡ú❞♦ s✉✜❝✐❡♥t❡ ♣❛r❛ r❡s♦❧✈❡r♠♦s q✉❛❧q✉❡r s✐st❡♠❛ ❧✐♥❡❛r✮✳

✸✳✻ ❈❧❛ss✐✜❝❛çã♦ ❞❡ ✉♠ ❙✐st❡♠❛ ▲✐♥❡❛r

❈❧❛ss✐✜❝❛♠♦s ♦s s✐st❡♠❛s ❧✐♥❡❛r❡s ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s✳ ❖ s✐st❡♠❛ s❡rá✿

• ♣♦ssí✈❡❧ ✭♦✉ ❝♦♠♣❛tí✈❡❧✮ ❡ ❞❡t❡r♠✐♥❛❞♦ q✉❛♥❞♦ ❛❞♠✐t✐r ❛♣❡♥❛s ✉♠❛ s♦❧✉çã♦❀

• ♣♦ssí✈❡❧ ❡ ✐♥❞❡t❡r♠✐♥❛❞♦ q✉❛♥❞♦ ♣♦ss✉✐r ✐♥✜♥✐t❛s s♦❧✉çõ❡s❀ • ✐♠♣♦ssí✈❡❧ ✭♦✉ ✐♥❝♦♠♣❛tí✈❡❧✮ s❡ ♥ã♦ ♣♦ss✉✐r s♦❧✉çã♦✳

❙✐st❡♠❛s ▲✐♥❡❛r❡s ❈❛♣ít✉❧♦ ✸

✸✳✼ ❘❡s♦❧✉çã♦ ❞❡ ✉♠ ❙✐st❡♠❛ ▲✐♥❡❛r

P❛r❛ r❡s♦❧✈❡r ✉♠ s✐st❡♠❛ ❧✐♥❡❛r✱ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ❛ r❡❣r❛ ❞❡ ❈r❛♠❡r✱ ♦ ♠ét♦❞♦ ❞❡ ●❛✉ss ♦✉ ♦ ♠ét♦❞♦ ❞❡ ●❛✉ss✲❏♦r❞❛♥✳

◆❡st❡ tr❛❜❛❧❤♦✱ ✈❛♠♦s ✉t✐❧✐③❛r ❛ ❡❧✐♠✐♥❛çã♦ ❣❛✉ss✐❛♥❛ ♣❛r❛ ❡♥❝♦♥tr❛r♠♦s ❛ s♦✲ ❧✉çã♦ ❞♦s s✐st❡♠❛s ❧✐♥❡❛r❡s ♣r♦♣♦st♦s✳

✸✳✼✳✶ ❊❧✐♠✐♥❛çã♦ ❣❛✉ss✐❛♥❛

❆ ❡❧✐♠✐♥❛çã♦ ❣❛✉ss✐❛♥❛ ✭♦✉ ❡s❝❛❧♦♥❛♠❡♥t♦✮ ❝♦♥s✐st❡ ❡♠ ♦r❣❛♥✐③❛r ♦ s✐st❡♠❛ ♥❛s s❡❣✉✐♥t❡s ❢♦r♠❛s✿

x+ 3y = 4 0x+ y = 1 ❀

 



x+ 2y₋z = 2 0x+ 5y+z = 1 0x+ 0y₋z= 7

❀

   

  

2x₋ y+ 5z+ 2w= 4 0x+ 3y+ 8z₋2w = 1 0x+ 0y₋ z₋3w= 0 0x+ 0y+ 0z+ 4w=₋8

.

◆♦t❡ q✉❡✱ ♥❡st❡s ❡①❡♠♣❧♦s✱ ♥❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ❛♣❛r❡❝❡♠ t♦❞❛s ❛s ✐♥❝ó❣♥✐t❛s✱ ♥❛ s❡❣✉♥❞❛✱ ③❡r❛♠ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ①✱ ♥❛ t❡r❝❡✐r❛✱ q✉❛♥❞♦ ❤á✱ ❛♣❛r❡❝❡ ③❡r♦ ♥♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ②✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳

❊ss❡ ♠ét♦❞♦ tr❛♥s❢♦r♠❛ ♦ s✐st❡♠❛ ❞❛❞♦ ♥✉♠ ♦✉tr♦ ❡q✉✐✈❛❧❡♥t❡ ❛♦ ♣r✐♠❡✐r♦✳ ❯s❛✲ r❡♠♦s ❛s três tr❛♥s❢♦r♠❛çõ❡s ❡❧❡♠❡♥t❛r❡s s♦❜r❡ ❛s ♠❛tr✐③❡s ♣❛r❛ ❝❤❡❣❛r ♥❡st❡ s✐st❡♠❛ ❡q✉✐✈❛❧❡♥t❡✿

• tr♦❝❛r ❛s ♣♦s✐çõ❡s ❞❡ ❞✉❛s ❧✐♥❤❛s✱ ✐st♦ é✱ Li _↔Lj✳

• ♠✉❧t✐♣❧✐❝❛r ✉♠❛ ❞❛ ❧✐♥❤❛s ♣♦r ✉♠ ♥ú♠❡r♦ ♥ã♦ ♥✉❧♦✱ ✐st♦ é✱ Li _↔k.Lj✳

• ♠✉❧t✐♣❧✐❝❛r ❛ ❧✐♥❤❛ i ♣♦r ✉♠ ♥ú♠❡r♦ r❡❛❧ ♥ã♦ ♥✉❧♦ ❡ ❛❞✐❝✐♦♥❛r ♦ r❡s✉❧t❛❞♦ ❛ ♦✉tr❛ ❧✐♥❤❛ j✱ ✐st♦ é✱Li _↔k.Li+Lj✳

P❛r❛ ❛♣❧✐❝❛r♠♦s ❡ss❛ ❡❧✐♠✐♥❛çã♦ ❣❛✉ss✐❛♥❛✱ ✈❛♠♦s ♣r♦❝❡❞❡r ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

• t♦r♥❛r✱ ❝❛s♦ ♥ã♦ s❡❥❛✱ ♦ ❝♦❡✜❝✐❡♥t❡ a11 ✐❣✉❛❧ ❛ ✶✳

• t♦r♥❛r ♥✉❧♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ x1 ♥❛s ❡q✉❛çõ❡s ❧♦❝❛❧✐③❛❞❛s ❛❜❛✐①♦ ❞❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦❀

• ❢❛③❡r✱ ❝❛s♦ s❡❥❛ ♥❡❝❡ssár✐♦✱ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ x2 ♥❛ s❡❣✉♥❞❛ ❡q✉❛çã♦ ✐❣✉❛❧ ❛ ✶❀

• r❡♣❡t✐r ♦ ♣r♦❝❡❞✐♠❡♥t♦ ❛❝✐♠❛ ♣❛r❛ a22♥❛ s❡❣✉♥❞❛ ❡q✉❛çã♦ ❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ x2 ♥❛s ❡q✉❛çõ❡s s❡❣✉✐♥t❡s❀ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳

• ❉❛ ú❧t✐♠❛ ❡q✉❛çã♦✱ ❞❡✈❡r❡♠♦s ❡♥❝♦♥tr❛r ♦ ✈❛❧♦r ❞❛ ✐♥❝ó❣♥✐t❛ xn✳ ❊ ❢❛③❡♥❞♦

✉♠❛ r❡tr♦✲s✉❜st✐t✉✐çã♦ ✭s✉❜st✐t✉✐çã♦ ♥❛ ♣❡♥ú❧t✐♠❛ ❡q✉❛çã♦✮✱ ♦❜t❡r❡♠♦s xn−1✳ ❖s ✈❛❧♦r❡s ❞❡ xn❡xn−1 s✉❜st✐t✉í❞♦s ♥❛ ❛♥t❡✲♣❡♥ú❧t✐♠❛ ❡q✉❛çã♦ ♥♦s ♣❡r♠✐t❡♠ ❡♥❝♦♥tr❛r ♦ ✈❛❧♦r xn−2✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱ ❡♥❝♦♥tr❛r❡♠♦s ♦s ✈❛❧♦r❡s ❞❡ t♦❞❛s ❛s ✐♥❝ó❣♥✐t❛s✳

❙✐st❡♠❛s ▲✐♥❡❛r❡s ❈❛♣ít✉❧♦ ✸

❱❛♠♦s r❡s♦❧✈❡r ♦ s✐st❡♠❛

4x₋3y=₋4 2x+ 4y= 10 .

▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ s❡❣✉♥❞❛ ❡q✉❛çã♦ ♣♦r 1

2 ❡ tr♦❝❛♥❞♦ ❛ ♣♦s✐çã♦ ❝♦♠ ❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦✱

♦❜t❡♠♦s✿

x+ 2y= 5 4x₋3y=₋4 .

❆❣♦r❛✱ ✈❛♠♦s ❡❧✐♠✐♥❛r ❛ ✐♥❝ó❣♥✐t❛ x ♥❛ s❡❣✉♥❞❛ ❧✐♥❤❛✳ ❇❛st❛ ♠✉❧t✐♣❧✐❝❛r♠♦s ❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ♣♦r −4❡ ❛❞✐❝✐♦♥❛r♠♦s ♦ r❡s✉❧t❛❞♦ ❝♦♠ ❛ s❡❣✉♥❞❛ ❡q✉❛çã♦✱ ❛ss✐♠✿

x+y= 5

0x₋11y=₋22 .

❉❛ s❡❣✉♥❞❛ ❡q✉❛çã♦ t❡♠♦s q✉❡✿

11y=₋22

y= 2.

❙✉❜st✐t✉✐♥❞♦ y ♣♦r2 ♥❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦✱ ♦❜t❡♠♦s✿

x+ 2.(2) = 5

x+ 4 = 5

x= 1. P♦rt❛♥t♦✱ ❛ s♦❧✉çã♦ ♣r♦❝✉r❛❞❛ é S=_{(1,2)_}✳

P♦❞❡♠♦s r❡s♦❧✈❡r s✐st❡♠❛s ❧✐♥❡❛r❡s ✉s❛♥❞♦ ❛♣❡♥❛s ♦s ❝♦❡✜❝✐❡♥t❡s✱ ✈❡❥❛♠♦s ♦ ❡①❡♠♣❧♦ ❛ ❛❜❛✐①♦✿

 



x+ 2y+ 4z = 5 2x₋y+ 2z = 8 3x₋3y₋z = 7

.

▲♦❣♦✱ ❡s❝r❡✈❡♥❞♦ ❛ ♠❛tr✐③ ❝♦♠♣❧❡t❛ ❡ ✉t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ❞❛ ❡❧✐♠✐♥❛çã♦ ❣❛✉ss✐❛♥❛✱

t❡♠♦s✿ _



1 2 4 5 2 ₋1 2 8 3 ₋3 ₋1 7



.