❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠
▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧
❘❡❝♦♥str✉✐♥❞♦ ♦s ❉❡t❡r♠✐♥❛♥t❡s✿ ❯♠❛ ♣r♦♣♦st❛
♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦
❍❛❧❧②s♦♥ ●♦♠❡s ▼♦♥s♦r❡s
❇r❛sí❧✐❛
❍❛❧❧②s♦♥ ●♦♠❡s ▼♦♥s♦r❡s
❘❡❝♦♥str✉✐♥❞♦ ♦s ❉❡t❡r♠✐♥❛♥t❡s✿
❯♠❛ ♣r♦♣♦st❛ ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡✳ ❖r✐❡♥t❛❞♦r✿ Pr♦❢❛ ❉r❛ ❆❧✐♥❡ P✐♥t♦
❇r❛sí❧✐❛
Ficha catalográfica elaborada automaticamente, com os dados fornecidos pelo(a) autor(a)
G755r
GOMES MONSORES, HALLYSON
RECONSTRUINDO OS DETERMINANTES: UMA PROPOSTA PARA O ENSINO MÉDIO / HALLYSON GOMES MONSORES; orientador ALINE PINTO. -- Brasília, 2015.
100 p.
Dissertação (Mestrado - Mestrado Profissional em Matemática) -- Universidade de Brasília, 2015.
❆❣r❛❞❡❝✐♠❡♥t♦s
❘❡s✉♠♦
❆ ♣❛❧❛✈r❛ ✬❞❡t❡r♠✐♥❛♥t❡✬ ❡ s❡✉ s✐❣♥✐✜❝❛❞♦ ❣❡r❛❧♠❡♥t❡ sã♦ ♠❛❧ ❝♦♠♣r❡❡♥❞✐❞❛s ♣❡❧♦s ❛❧✉♥♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❆ ♥❛t✉r❡③❛ ❛❜str❛t❛ ❞❡st❡ t❡r♠♦ ❡ ❛s ✐♥❛❞❡q✉❛❞❛s té❝♥✐❝❛s tr❛❞✐❝✐♦♥❛✐s ❞❡ ❡♥s✐♥♦ ♣♦❞❡♠ ❡①♣❧✐❝❛r ❡st❛ s✐t✉❛çã♦✳ ❉❡♣♦✐s ❞❡ ❡s❝❧❛r❡❝❡r ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ✬❞❡t❡r♠✐♥❛♥t❡✬✱ ❡st❛ ❞✐ss❡rt❛çã♦ ✈✐s❛ ♣r♦✈❛r ❛ s✉❛ ❡①✐stê♥❝✐❛✳ ▼❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ♥ós ♠♦str❛♠♦s q✉❡ ❡①✐st❡ ❛ ❢✉♥çã♦ ❞❡t❡r♠✐♥❛♥t❡ ♣❛r❛ ♠❛tr✐③❡s ❞❡ q✉❛❧q✉❡r ♦r❞❡♠n✳
❆❧é♠ ❞✐ss♦✱ ❛♣r❡s❡♥t❛♠♦s ♠ét♦❞♦s ♣❛r❛ ❢❛❝✐❧✐t❛r ♦ ❝á❧❝✉❧♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ♥♦s ❝❛s♦s ♠❛✐s ❝♦♠♣❧❡①♦s✳ ❋✐♥❛❧♠❡♥t❡✱ ♣r♦♣♦♠♦s té❝♥✐❝❛s ♣rát✐❝❛s ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❡♠ ❝♦♥t❡①t♦s ❡s❝♦❧❛r❡s✳
P❛❧❛✈r❛s✲❝❤❛✈❡
❉❡t❡r♠✐♥❛♥t❡✱ ❛❜str❛çã♦✱ ❡♥t❡♥❞✐♠❡♥t♦✳
❆❜str❛❝t
❚❤❡ ✇♦r❞ ✬❞❡t❡r♠✐♥❛♥t✬ ❛♥❞ ✐ts ♠❡❛♥✐♥❣ ❛r❡ ♦❢t❡♥ ♣♦♦r❧② ✉♥❞❡rst♦♦❞ ❜② s❝❤♦♦❧ st✉✲ ❞❡♥ts✳ ❚❤❡ ❛❜str❛❝t ♥❛t✉r❡ ♦❢ t❤✐s t❡r♠ ❛♥❞ ✉♥s✉✐t❛❜❧❡ tr❛❞✐t✐♦♥❛❧ t❡❛❝❤✐♥❣ t❡❝❤♥✐q✉❡s ♠❛② ❡①♣❧❛✐♥ t❤✐s s✐t✉❛t✐♦♥✳ ❆❢t❡r ❝❧❛r✐❢②✐♥❣ t❤❡ ♠❡❛♥✐♥❣ ♦❢ ✬❞❡t❡r♠✐♥❛♥t✬✱ t❤✐s ❞✐ss❡r✲ t❛t✐♦♥ s❡❡❦s t♦ ♣r♦✈❡ ✐ts ❡①✐st❡♥❝❡✳ ▼♦r❡ s♣❡❝✐✜❝❛❧❧②✱ ✇❡ s❤♦✇ t❤❛t t❤❡ ❞❡t❡r♠✐♥❛♥t ❢✉♥❝t✐♦♥ ❡①✐sts ❢♦r ♠❛tr✐❝❡s ♦❢ ❛♥② ♦r❞❡rn✳ ▼♦r❡♦✈❡r✱ ✇❡ ♣r❡s❡♥t ♠❡t❤♦❞s t♦ ❢❛❝✐❧✐t❛t❡
t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ❞❡t❡r♠✐♥❛♥ts ✐♥ ♠♦r❡ ❝♦♠♣❧❡① ❝❛s❡s✳ ❋✐♥❛❧❧②✱ ✇❡ ♣r♦♣♦s❡ ♣r❛❝t✐❝❛❧ t❡❝❤♥✐q✉❡s ❢♦r s♦❧✈✐♥❣ s②st❡♠s ♦❢ ❡q✉❛t✐♦♥ ✐♥ s❝❤♦♦❧ ❝♦♥t❡①ts✳
❑❡②✇♦r❞s
❉❡t❡r♠✐♥❛♥t✱ ❛❜str❛❝t✐♦♥✱ ✉♥❞❡rst❛♥❞✐♥❣✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ r❡t❛s ❝♦♥❝♦rr❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷ r❡t❛s ❝♦✐♥❝✐❞❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✸ r❡t❛s ♣❛r❛❧❡❧❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✹ P❧❛♥♦s ❝✉❥❛ ✐♥t❡rs❡❝çã♦ é ✉♠ ♣♦♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✺ P❧❛♥♦s ❝✉❥❛ ✐♥t❡rs❡❝çã♦ é ✉♠❛ r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✻ P❧❛♥♦s ❝✉❥❛ ✐♥t❡rs❡❝çã♦ é ✉♠❛ r❡t❛ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✼ P❧❛♥♦s ❝✉❥❛ ✐♥t❡rs❡❝çã♦ é ✉♠ ♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✽ P❧❛♥♦s s❡♠ ✐♥t❡rs❡❝çã♦ tr✐♣❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✾ P❧❛♥♦s s❡♠ ✐♥t❡rs❡❝çã♦ tr✐♣❧❛ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✶✵ P❧❛♥♦s s❡♠ ✐♥t❡rs❡❝çã♦ tr✐♣❧❛ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
❙✉♠ár✐♦
✶ ❖r✐❣❡♠ ❡ ❡✈♦❧✉çã♦ ❤✐stór✐❝❛ ✶✺
✷ ❘❡❝♦♥str✉✐♥❞♦ ♦ ❉❡t❡r♠✐♥❛♥t❡ ✷✵
✷✳✶ ❉❡t❡r♠✐♥❛♥t❡ ❞❡ ♦r❞❡♠ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✶✳✶ ❈❛s♦s ❞♦ ❙✐st❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✶✳✷ ■♥❞❡①❛♥❞♦ ♦s ❊❧❡♠❡♥t♦s ❞❛ ▼❛tr✐③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✶✳✸ ❆♥❛❧♦❣✐❛ ●❡♦♠étr✐❝❛ ❞♦s ❈❛s♦s ❞♦ ❙✐st❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷ ❉❡t❡r♠✐♥❛♥t❡ ❞❡ ♦r❞❡♠ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✷✳✶ ❈❛s♦s ❞♦ ❙✐st❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✷✳✷ ■♥❞❡①❛♥❞♦ ♦s ❊❧❡♠❡♥t♦s ❞❛ ▼❛tr✐③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✷✳✸ ❘❡❣r❛ ❞❡ ❙❛rr✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✷✳✹ ❆♥❛❧♦❣✐❛ ●❡♦♠étr✐❝❛ ❞♦s ❈❛s♦s ❞♦ ❙✐st❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✸ ▼❛tr✐③❡s ✹✻
✸✳✶ ❚✐♣♦s ❞❡ ▼❛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 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✹ ❋✉♥❞❛♠❡♥t❛çã♦ ❚❡ór✐❝❛ ❞♦ ❉❡t❡r♠✐♥❛♥t❡ ✺✺
✹✳✶ ❋✉♥çõ❡s ❉❡t❡r♠✐♥❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✹✳✷ ❯♥✐❝✐❞❛❞❡ ❞❛ ❋✉♥çã♦ ❉❡t❡r♠✐♥❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✹✳✸ Pr♦♣r✐❡❞❛❞❡s ❞♦s ❉❡t❡r♠✐♥❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✹✳✹ ❆❜❛✐①❛♠❡♥t♦ ❞❡ ♦r❞❡♠ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛♥t❡ ✭❘❡❣r❛ ❞❡ ❈❤✐ó✮ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✹✳✺ ❘❡❣r❛ ❞❡ ❈r❛♠♠❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸
✺ ❆t✐✈✐❞❛❞❡s Pr♦♣♦st❛s ✽✻ ✺✳✶ ❆t✐✈✐❞❛❞❡s s♦❜r❡ ♠❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✺✳✷ ❆t✐✈✐❞❛❞❡s s♦❜r❡ ❉❡t❡r♠✐♥❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵
■♥tr♦❞✉çã♦
❉❡ ❢♦r♠❛ ❣❡r❛❧✱ ❛♦ ♣❛ss❛r♠♦s ♣❡❧♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♥♦s ❞❡♣❛r❛♠♦s ❝♦♠ ❛s ♥♦çõ❡s ❞❡ ♠❛tr✐③❡s ❡ ❞❡t❡r♠✐♥❛♥t❡s ❞❡ ♠❛♥❡✐r❛ ♠❡❝❛♥✐③❛❞❛ ❡ s❡♠ ❛ ♣r❡♦❝✉♣❛çã♦ ❞❡ ✉♠❛ ❝♦♥t❡①t✉❛❧✐③❛çã♦ ❤✐stór✐❝❛ ❞♦ s✉r❣✐♠❡♥t♦ ❞♦ ❛ss✉♥t♦ ❜❡♠ ❝♦♠♦ ✉♠❛ ❡①♣❧✐❝❛çã♦ ♣rát✐❝❛ ❞♦s ♦❜❥❡t✐✈♦s ♣❛r❛ ♦s q✉❛✐s ❢♦r❛♠ ❝r✐❛❞♦s✳
●❡r❛❧♠❡♥t❡✱ ❛♣ós ✉♠❛ ❜r❡✈❡ ❡ ♥ã♦ ♠✉✐t♦ ❡✜❝✐❡♥t❡ ❝♦♥❝❡✐t✉❛çã♦ ❛❧❣é❜r✐❝❛✱ ❛s ♠❛✲ tr✐③❡s sã♦ ❡①♣♦st❛s ❛♦ ❛❧✉♥♦ ❝♦♠♦ t❛❜❡❧❛s ❞❡ ✈❛❧♦r❡s ❛❣r✉♣❛❞❛s ❡♠ ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s ❡ ♦s ❞❡t❡r♠✐♥❛♥t❡s sã♦ ♠♦str❛❞♦s ❝♦♠♦ ✈❛❧♦r❡s r❡❛✐s ❛ss♦❝✐❛❞♦s ❛ ♠❛tr✐③❡s q✉❛❞r❛❞❛s ❡ ♦❜t✐❞♦s ♣♦r ✉♠ ♠ét♦❞♦ ❛❧❣♦r✐t♠✐③❛❞♦✳ ◗✉❛s❡ s❡♠♣r❡✱ ♦ ❛❧✉♥♦ s❡ ❞❡♣❛r❛ ❝♦♠ ❞ú✈✐❞❛s q✉❡ ♥ã♦ sã♦ r❡s♣♦♥❞✐❞❛s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♦ q✉❡ ❛❝❛❜❛ ♣♦r ♠❡❝❛♥✐③❛r ♦ ❡♥s✐♥♦ ❡ ♥❛ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s ❞❡s❡st✐♠✉❧❛ ♦ ✐♥t❡r❡ss❡ ❞♦s ❛❧✉♥♦s ❛ ❝❡r❝❛ ❞♦ ♦❜❥❡t♦ ❡st✉❞❛❞♦✳
P❡r❣✉♥t❛s ❡ss❡♥❝✐❛✐s ❛♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ ❛ss✉♥t♦ ❝♦♠♦ ✏❖ q✉❡ é ♦ ❞❡t❡r♠✐♥❛♥t❡❄✑✱ ✏❖ q✉❡ é ✉♠❛ ♠❛tr✐③❄✑✱ ✏P❛r❛ q✉❡ s❡r✈❡ ♦ ❞❡t❡r♠✐♥❛♥t❡❄✑✱ ✏P♦r q✉❡ só ♠❛tr✐③❡s q✉❛✲ ❞r❛❞❛s ♣♦ss✉❡♠ ❞❡t❡r♠✐♥❛♥t❡❄✑✱ ❡♥tr❡ ♦✉tr❛s✱ ♥ã♦ sã♦ r❡s♣♦♥❞✐❞❛s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ♦ q✉❡ ❢❛③ ❝♦♠ q✉❡ ♦ ❛❧✉♥♦ ❛❜s♦r✈❛ ❛♣❡♥❛s ❛ ❛❧❣♦r✐t♠✐③❛çã♦ ♠❛s ♥ã♦ ❡♥t❡♥❞❛ ♦ s❡♥t✐❞♦ ❡ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ✉t✐❧✐③❛❞♦✱ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♦s ❛❧✉♥♦s r❡♣❧✐❝❛♠ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ♠❡❝❛♥✐③❛❞♦ s❡♠ s❛❜❡r ♦ q✉❡ ❡stã♦ ❢❛③❡♥❞♦ ♥❡♠ ♣❛r❛ q✉❡ ♦ ❢❛③❡♠✳
◆❡st❡ tr❛❜❛❧❤♦✱ ❛❜♦r❞❛r❡♠♦s ♦ ❛ss✉♥t♦ ✏❉❡t❡r♠✐♥❛♥t❡s✑ tr❛③❡♥❞♦ ❛ ❧✉③ s✉❛ ♦r✐❣❡♠ ❣r❛❞✉❛❧ ♥❛ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛ ♠♦str❛♥❞♦ ❛ss✐♠ ❞❡ ❢♦r♠❛ ♠❛✐s ❝❧❛r❛ ♦ q✉❡ é ♦ ❞❡t❡r♠✐♥❛♥t❡✱ ❝♦♠♦ ❡❧❡ s✉r❣✐✉✱ ❝♦♠♦ ❡✈♦❧✉✐✉ ❛té ♦s ❞✐❛s ♠♦❞❡r♥♦s✱ q✉❛✐s ❡r❛♠ ♦s ♦❜❥❡t✐✈♦s ✐♥✐❝✐❛✐s ❛♦ s❡ ❡st✉❞❛r ❡ss❡ ❛ss✉♥t♦ ❡ ❡♠ q✉❡ ár❡❛s ❞❛ ❝✐ê♥❝✐❛ ♠❛t❡♠át✐❝❛ ❡❧❡ é ✉t✐❧✐③❛❞♦ ❤♦❥❡✳
P♦r ♠❡✐♦ ❞♦ ♣r♦❝❡ss♦ ❞❡ r❡❝♦♥str✉çã♦ ❞❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛✲ r❡s ♠♦str❛r❡♠♦s ❞❡ ❢♦r♠❛ ❛❧❣é❜r✐❝❛ ❣❡♥❡r❛❧✐③❛❞❛ ♦ q✉❡ é ♦ ❞❡t❡r♠✐♥❛♥t❡ ♥♦s ❝❛s♦s ❞❡ ✷❛ ❡ ✸❛ ♦r❞❡♠✱ ❝♦♠♦ ❡❧❡ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ❡ ❝♦♠♦ ❡ss❡s ❞❡t❡r♠✐♥❛♥t❡s sã♦ ✐♠♣♦rt❛♥✲
t❡s ♥❛ ❢❛❝✐❧✐t❛çã♦ ❞❛ r❡s♦❧✉çã♦ ❞❡ss❡s s✐st❡♠❛s✳ ❚r❛t❛r❡♠♦s ❡ss❡ ♣r♦❝❡ss♦ ❝♦♠♦ ✉♠❛ ♣r♦♣♦st❛ ❛❧t❡r♥❛t✐✈❛ ❞❡ ✐♥tr♦❞✉çã♦ ❛♦ ❡st✉❞♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ❡♠ ❞❡tr✐♠❡♥t♦ ❛♦ ♣r♦✲
❝❡ss♦ tr❛❞✐❝✐♦♥❛❧ ❞❡ ❡♥s✐♥♦ ❛♣❧✐❝❛❞♦✳ ❖ ♣r♦❝❡ss♦ q✉❡ ✉t✐❧✐③❛r❡♠♦s ♥❛ r❡❝♦♥str✉çã♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ❞❡ ✷❛ ❡ ✸❛ ♦r❞❡♠✱ ❛♣❡s❛r ❞❡ t♦t❛❧♠❡♥t❡ ❛❧❣é❜r✐❝♦ é ❜❡♠ ❜ás✐❝♦✱ ❝♦♥s✐s✲
t✐♥❞♦ s♦♠❡♥t❡ ❡♠ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❡q✉❛çõ❡s ❡ ❡❧✐♠✐♥❛çã♦ ❞❡ ✐♥❝ó❣♥✐t❛s✱ ❧♦❣♦ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ❡♠ s❛❧❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ♣❛r❛ ✐♥tr♦❞✉③✐r ♦ ❡st✉❞♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s✱ ♣♦✐s ♣❛r❛ ❡♥t❡♥❞ê✲❧♦ ❜❛st❛ q✉❡ ♦ ❛❧✉♥♦ ❡st❡❥❛ ❢❛♠✐❧✐❛r✐③❛❞♦ ❝♦♠ ❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ♣❡❧♦ ♠ét♦❞♦ ❞❛ ❡❧✐♠✐♥❛çã♦ ❞❡ ✐♥❝ó❣♥✐t❛s✳
❈❛❜❡ s❛❧✐❡♥t❛r q✉❡ ♣❛r❛ ❞❡t❡r♠✐♥❛♥t❡s ❞❡ ♦r❞❡♠ ♠❛✐♦r q✉❡ ✸✱ ♦ s✐st❡♠❛ ❞❡ r❡❝♦♥s✲ tr✉çã♦ ❛❧❣é❜r✐❝❛ ♣❡❧❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❝♦♠❡ç❛ ❛ ✜❝❛r ♠✉✐t♦ tr❛❜❛❧❤♦s♦ ❡ ✐♠♣r♦❞✉t✐✈♦✱ ✉t✐❧✐③❛♥❞♦ ❧❡tr❛s ❞❡♠❛✐s ♣❛r❛ r❡♣r❡s❡♥t❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❡ ♥❡✲ ❝❡ss✐t❛♥❞♦ ❞❡ ♠✉✐t❛s ♦♣❡r❛çõ❡s ❛❧❣é❜r✐❝❛s ♣❛r❛ ❝♦♥❝❧✉í✲❧♦✳ ❈♦♠♦ ❛ ♠❛✐♦r✐❛ ❞♦s s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❡st✉❞❛❞♦s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ sã♦ ❞❡ ✷❛ ♦✉ ❞❡ ✸❛ ♦r❞❡♠✱ t♦r♥❛✲s❡ ❞❡s♥❡❝❡ssá✲
r✐♦ ❣❡♥❡r❛❧✐③❛r ♦ ❞❡t❡r♠✐♥❛♥t❡ ♣❛r❛ ❛s ❞❡♠❛✐s✱ ♣♦ré♠ ❝♦♠ ♦ ✏❚❡♦r❡♠❛ ❞❡ ▲❛♣❧❛❝❡✑ q✉❡ ❡st✉❞❛r❡♠♦s t❛♠❜é♠ ♥❡st❡ tr❛❜❛❧❤♦✱ t♦r♥❛✲s❡ ♣♦ssí✈❡❧ ❝❛❧❝✉❧❛r ♦ ❞❡t❡r♠✐♥❛♥t❡ ❛♣❧✐❝❛❞♦ s♦❜r❡ ♠❛tr✐③❡s ❞❡ q✉❛❧q✉❡r ♦r❞❡♠✳
P❛ss❛♥❞♦ ❛ ✉♠ ♥í✈❡❧ ♠❛✐s ❡❧❡✈❛❞♦ ❞♦ ❡st✉❞♦✱ ❡♥✉♥❝✐❛r❡♠♦s ❡ ♣r♦✈❛r❡♠♦s ❛ ❡①✐s✲ tê♥❝✐❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ❞❡♥♦♠✐♥❛❞❛ ❝♦♠♦ ✏❋✉♥çã♦ ❉❡t❡r♠✐♥❛♥t❡✑ q✉❡ é ❛ss♦❝✐❛❞❛ ❛ ♠❛tr✐③❡s ❞❡ ♦r❞❡♠ n✱ ❡♠ s❡❣✉✐❞❛ ♥♦s ❞❡❞✐❝❛r❡♠♦s ❛ ♠♦str❛r q✉❡ ❡ss❛ ❢✉♥çã♦ é ú♥✐❝❛✱
♦✉ s❡❥❛ q✉❡ só ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ❞❡t❡r♠✐♥❛♥t❡ ❛♣❧✐❝❛❞❛ s♦❜r❡ ♠❛tr✐③❡s ❞❡ ♦r❞❡♠ n ❡
♣♦r ú❧t✐♠♦ ❛❜♦r❞❛r❡♠♦s ❞✐✈❡rs❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❚❡♦r❡♠❛s q✉❡ ❢❛❝✐❧✐t❛♠ ♦ ❝á❧❝✉❧♦ ❡ ♦ tr❛❜❛❧❤♦ ❝♦♠ ❞❡t❡r♠✐♥❛♥t❡s ❞❡ q✉❛❧q✉❡r ♦r❞❡♠✳
❈❛♣ít✉❧♦ ✶
❖r✐❣❡♠ ❡ ❡✈♦❧✉çã♦ ❤✐stór✐❝❛
❙❡❣✉♥❞♦ ❇♦②❡r ✭❬✸❪✱ ♣♣✳ ✶✸✸✲✶✸✺✮ ❡ ■❡③③✐ ❡ ❍❛③③❛♥ ✭❬✶❪✱ ♣♣✳ ✶✷✺✲✶✷✻✮✱ ❛♣❡s❛r ❞❡ ❛❧❣✉✲ ♠❛s ❞✐✈❡r❣ê♥❝✐❛s ❡♥tr❡ ♦s ❤✐st♦r✐❛❞♦r❡s ❞❛ ♠❛t❡♠át✐❝❛✱ ❡♠ ✉♠ ❛s♣❡❝t♦ ❡❧❡s ❝♦♥✈❡r❣❡♠✱ ♦ ❡st✉❞♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s t❡✈❡ s✉❛ ♦r✐❣❡♠ ♠❛✐s r❡♠♦t❛ ✐♥✐❝✐❛❞❛ ♣❡❧❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ r❡s♦❧✉çã♦ ❞♦s s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s✳ ❋♦r❛♠ ❡♥❝♦♥tr❛❞♦s r❡❣✐str♦s ❞❡ t❛❜✉❧❡t❛s ❡♠ ❛r❣✐❧❛ ❜❛❜✐❧ô♥✐❝❛s ❞❛t❛❞❛s ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✸✵✵ ❛♥♦s ❛✳❝✳ ♦♥❞❡ ❡♥❝♦♥tr❛♠♦s ♦s ♣r✐♠❡✐r♦s r❡❣✐str♦s ❝♦♥❤❡❝✐❞♦s ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❞❡ ❞✉❛s ❡q✉❛çõ❡s ❡ ❞✉❛s ✐♥❝ó❣♥✐t❛s s❡♥❞♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s✳
❆✐♥❞❛ ♥❛ ❛♥t✐❣✉✐❞❛❞❡ ♦s ♦r✐❡♥t❛✐s ❝♦♠❡ç❛r❛♠ ❛ ❡st✉❞❛r ❡ss❡ t✐♣♦ ❞❡ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s✳ ❖ ❝❤✐♥❡s❡s✱ ❡s♣❡❝✐❛❧✐st❛s ❡♠ ❞✐❛❣r❛♠❛s ❡ q✉❛❞r❛❞♦s ♠á❣✐❝♦s✱ r❡♣r❡s❡♥t❛✈❛♠ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❛tr❛✈és ❞❡ s❡✉s ❝♦❡✜❝✐❡♥t❡s ❡s❝r✐t♦s ❡♠ ❜❛rr❛s ❞❡ ❜❛♠❜✉ s♦❜r❡ ♦s q✉❛❞r❛❞♦s ❞❡ ✉♠ t❛❜✉❧❡✐r♦✳
❆♣r♦①✐♠❛❞❛♠❡♥t❡ ♥♦ sé❝✉❧♦ ■■ ❛✳❝✳ ❢♦✐ ♣✉❜❧✐❝❛❞♦ ♥❛ ❈❤✐♥❛✱ ♦ ❧✐✈r♦ ✏❖s ♥♦✈❡ ❝❛♣ít✉✲ ❧♦s s♦❜r❡ ❛ ❛rt❡ ♠❛t❡♠át✐❝❛✑ q✉❡ ♣r♦♣✉♥❤❛ ❝❡r❝❛ ❞❡ ✷✹✻ ♣r♦❜❧❡♠❛s s♦❜r❡ ♠❡♥s✉r❛çã♦ ❞❡ t❡rr❛s✱ ❛❣r✐❝✉❧t✉r❛✱ s♦❝✐❡❞❛❞❡s✱ ❡♥❣❡♥❤❛r✐❛✱ ✐♠♣♦st♦s✱ ❝á❧❝✉❧♦s✱ s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s ❞♦s tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s ❛ s❡r❡♠ r❡s♦❧✈✐❞♦s ♣♦r ♠❡✐♦ ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s✳ ❉❡ ❢♦r♠❛ q✉❛s❡ q✉❡ ❛❧❣♦r✐t♠✐③❛❞❛ ♦ ❧✐✈r♦ r❡s♦❧✈✐❛ ♦s s✐st❡♠❛s ♣♦r ❡❧✐♠✐♥❛çã♦✱ ❛♥✉❧❛♥❞♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ✐♥❝ó❣♥✐t❛s ❛tr❛✈és ❞❡ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s ♠✉✐t♦ s❡♠❡❧❤❛♥t❡s ❛♦ q✉❡ ❢❛③❡♠♦s ❤♦❥❡ ❡♠ ❞✐❛ ❛♦ ❡s❝❛❧♦♥❛r ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ♣❡❧♦ ♣r♦❝❡ss♦ ❞❡ ●❛✉ss✳ ❯♠ ❞♦s ♣r✐♠❡✐r♦s ♣r♦❜❧❡♠❛s ❞❡st❡ ❧✐✈r♦ ❡ t❛❧✈❡③ ❛q✉❡❧❡ q✉❡ t❡♥❤❛ ❣❛♥❤❛❞♦ ♠❛✐♦r r❡♣❡r❝✉ssã♦ ❝♦♥t✐♥❤❛ ♦ s❡❣✉✐♥t❡ ❡♥✉♥❝✐❛❞♦✿
✏❍❛✈✐❛ três t✐♣♦s ❞❡ ♠✐❧❤♦✿ ❚rês ♣❛❝♦t❡s ❞♦ ♣r✐♠❡✐r♦ t✐♣♦✱ ❞♦✐s ❞♦ s❡❣✉♥❞♦ ❡ ✉♠ ❞♦ t❡r❝❡✐r♦✱ t♦t❛❧✐③❛♥❞♦ ✸✾ ✉♥✐❞❛❞❡s ❞❡ ♠✐❧❤♦✳ ❉♦✐s ♣❛❝♦t❡s ❞♦ ♣r✐♠❡✐r♦✱ três ♣❛❝♦t❡s
❞♦ s❡❣✉♥❞♦ ❡ ✉♠ ♣❛❝♦t❡ ❞♦ t❡r❝❡✐r♦✱ t♦t❛❧✐③❛♥❞♦ ✸✹ ✉♥✐❞❛❞❡s ❞❡ ♠✐❧❤♦ ❡✱ ✉♠ ♣❛❝♦t❡ ❞♦ ♣r✐♠❡✐r♦✱ ❞♦✐s ❞♦ s❡❣✉♥❞♦ ❡ três ❞♦ t❡r❝❡✐r♦✱ s♦♠❛♥❞♦ ✷✻ ✉♥✐❞❛❞❡s ❞❡ ♠✐❧❤♦✳ ❙❛❜❡♥❞♦ q✉❡ ♦s ♣❛❝♦t❡s ❞♦ ♠❡s♠♦ t✐♣♦ ❞❡ ♠✐❧❤♦✱ ❧❡✈❛♠ ❛ ♠❡s♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✉♥✐❞❛❞❡s✱ q✉❛♥t❛s ✉♥✐❞❛❞❡s ❞❡ ♠✐❧❤♦ ❝♦♥t❡♠ ✉♠ ♣❛❝♦t❡ ❞❡ ❝❛❞❛ t✐♣♦❄ ✑
❆ r❡s♦❧✉çã♦ ❞❛❞❛ ♣❡❧♦ ❧✐✈r♦ ❛❣r✉♣❛ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ✐♥❝ó❣♥✐t❛s ✭♣❛❝♦t❡s ❞❡ ❝❛❞❛ t✐♣♦ ❞❡ ♠✐❧❤♦✮ ♦r❣❛♥✐③❛❞♦s ❡♠ ✉♠❛ t❛❜❡❧❛ ♠✉✐t♦ ♣❛r❡❝✐❞❛ ❝♦♠ ♦ q✉❡ s❡r✐❛ ❤♦❥❡ ✉♠❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠4×3✱ ❝♦♠♦ ♣♦❞❡♠♦s ✈✐s✉❛❧✐③❛r ❛❜❛✐①♦✳ ❊♠ s❡❣✉✐❞❛ sã♦ ❢❡✐t❛s ❞✐✈❡rs❛s ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s q✉❡ ♦❜❥❡t✐✈❛♠ ③❡r❛r ♦s ❝♦❡✜❝✐❡♥t❡s ✉♠ ❛ ✉♠ ❡ ❛ss✐♠ ❝❤❡❣❛r ❛ ✉♠❛ ❡q✉❛çã♦ ♠❛✐s s✐♠♣❧❡s ❝♦♠ ✉♠❛ ú♥✐❝❛ ✐♥❝ó❣♥✐t❛✳ ❆♣ós ❡♥❝♦♥tr❛r ❛ s♦❧✉çã♦ ♣❛r❛ ❡ss❛ ✐♥❝ó❣♥✐t❛ ♦ s✐st❡♠❛ ✈❛✐ s❡♥❞♦ r❡s♦❧✈✐❞♦ s✉❜st✐t✉✐♥❞♦ ♦ ✈❛❧♦r ❡♥❝♦♥tr❛❞♦✱ ❡♥❝♦♥tr❛♥❞♦ ♦ ✈❛❧♦r ❞❛ ♣ró①✐♠❛ ✐♥❝ó❣♥✐t❛ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ ❖❜✈✐❛♠❡♥t❡ ❡ss❡ ♣r♦❝❡❞✐♠❡♥t♦ ❤♦❥❡ é ♦ q✉❡ ♠❛✐s s❡ ❛♣r♦①✐♠❛ ❞♦ ❡s❝❛❧♦♥❛♠❡♥t♦ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ●❛✉ss✳
1 2 3
2 3 2
3 1 1
26 34 39
⇒
0 0 3
0 5 2
36 1 1
99 24 39
❆♣❡s❛r ❞❡ ♦ ✐♥✐❝✐♦ ❞♦s ❡st✉❞♦s ❛❝❡r❝❛ ❞❡ss❡ ❛ss✉♥t♦ ❞❛t❛r ❞❡ ❛♥t❡s ❞❡ ❝r✐st♦✱ ❡st❡ ✜❝♦✉ ✐♥❡rt❡ ❡ s❡♠ ❡✈♦❧✉çã♦ ♠❛t❡r✐❛❧ ❡ s♦♠❡♥t❡ ♥♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ❳❱■■ q✉❡ ❛ ✐❞❡✐❛ ❞❡ ❞❡t❡r♠✐♥❛♥t❡ ❝♦♠❡ç♦✉ ❛ t♦♠❛r ❝♦r♣♦✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ ❢♦✐ ♥♦ tr❛❜❛❧❤♦ ❞♦ ♠❛✲ t❡♠át✐❝♦ ❥❛♣♦♥ês ❙❡❦✐ ❚❛❦❛❦❛③✉ ❑♦✇❛ ✭✶✻✹✷✲✶✼✵✽✮ ❡♠ ✶✻✽✸ q✉❡ ♦ ♠ét♦❞♦ ❝❤✐♥ês ❢♦✐ ❛♣r✐♠♦r❛❞♦ ❡ ❢♦✐ ♣❛❞r♦♥✐③❛❞♦ ✉♠❛ ❢♦r♠❛ ❞❡ r❡s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s✱ ♣♦ré♠ ❛♣❡♥❛s ♣❛r❛ ♦ ❝❛s♦ ❞❡ ❞✉❛s ❡q✉❛çõ❡s ❡ ❞✉❛s ✐♥❝ó❣♥✐t❛s✳ ❊ss❛ ❛❧❣♦r✐t♠✐③❛çã♦ ❞♦ ♠ét♦❞♦ ❝❤✐♥ês ✉s❛✈❛ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ✐♥❝ó❣♥✐t❛s ❞❛s ❡q✉❛çõ❡s ❞♦ s✐st❡♠❛ ❧✐♥❡❛r ❡♥❝♦♥tr❛♥❞♦ ❡s❝❛❧❛r❡s q✉❡ s♦❧✉❝✐♦♥❛✈❛♠ ❡st❡ s✐st❡♠❛✳ ❍♦❥❡ ❝♦♥❤❡❝❡♠♦s ❡ss❡s ❡s❝❛❧❛r❡s ❝♦♠♦ ❞❡t❡r♠✐♥❛♥t❡s ❞❛s ♠❛tr✐③❡s ♦❜t✐❞❛s ❝♦♠ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ❡q✉❛çõ❡s ❡ t❛♠❜é♠ ❝♦♠ ♦s t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s✳
❙❡❣✉♥❞♦ ❇♦②❡r ✭❬✸❪✱ ♣♣✳ ✷✼✾✲✷✽✵✮✱ ❢♦✐ ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ▲❡✐❜♥✐③ ✭✶✻✹✾✲✶✼✶✻✮ ♦ ♣r✐♠❡✐r♦ ♠❛t❡♠át✐❝♦ ❛ ❢❛③❡r r❡❢❡rê♥❝✐❛ ❛♦ ❡st✉❞♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ♥♦ ♦❝✐❞❡♥t❡ ❝❡r❝❛ ❞❡ ✶✵ ❛♥♦s ❞❡♣♦✐s ❞❛s ♣✉❜❧✐❝❛çõ❡s ❞❡ ❙❡❦✐ ❑♦✇❛ ♥♦ ❖r✐❡♥t❡✳ ❆♣❡s❛r ❞❛ ♠❛✐♦r ♣❛rt❡ ❞❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡ ▲❡✐❜♥✐③ ♣❛r❛ ❛ ♠❛t❡♠át✐❝❛ ✈❡rs❛r s♦❜r❡ ♦ ❝á❧❝✉❧♦✱ ♦s ❡st✉❞♦s ❞❡ ▲❡✐❜♥✐③ t❛♠❜é♠ ❢♦r❛♠ ❞✐r❡❝✐♦♥❛❞♦s ♥♦ ✐♥t✉✐t♦ ❞❡ ❛❧❣♦r✐t♠✐③❛r ❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ❞❡ três ❡q✉❛çõ❡s ❡ ❞✉❛s ✐♥❝ó❣♥✐t❛s ♣♦ré♠ ✉t✐❧✐③❛♥❞♦ ♥ã♦ ✉♠❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠
2× 2✱ ♠❛s ✉♠❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠ 3 ×3 ❢♦r♠❛❞❛ ♣❡❧♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ❡q✉❛çõ❡s ❡ ♣❡❧♦s t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❖ ♣r♦❝❡ss♦ ❞❡ ▲❡✐❜♥✐③ ❢♦✐ ♦ ♠❛✐s ♣ró①✐♠♦ ❞♦ q✉❡ s❡r✐❛ ❤♦❥❡ ♦ ❡st✉❞♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ❞❡ ♠❛tr✐③❡s ❞❡ ♦r❞❡♠ 3×3✳ ❋♦✐ t❛♠❜é♠ ▲❡✐❜♥✐③ ♦ ♣r✐♠❡✐r♦ ♠❛t❡♠át✐❝♦ ❛ ✐♥s❡r✐r í♥❞✐❝❡s ♣❛r❛ ♦s ❝♦❡✜❝✐❡♥t❡s✱ ❢❛t♦ ❝♦♠♣r♦✈❛❞♦ ❡♠ ❝❛rt❛s ❞❛t❛❞❛s ❞❡ ✶✻✾✸✱ ♦♥❞❡ ▲❡✐❜♥✐③ ❝♦♥t❛ ❛ ●✉✐❧❧❛✉♠❡ ❋r❛♥ç♦✐s ▲✬❍♦s♣✐t❛❧ ✭✶✻✻✶✲✶✼✵✹✮ q✉❡ ♦❝❛s✐♦♥❛❧♠❡♥t❡ ✉t✐❧✐③❛✈❛ ♥ú♠❡r♦s ✐♥❞✐❝❛♥❞♦ ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s ♣❛r❛ ✐♥❞❡①❛r ✉♠❛ ❝♦❧❡çã♦ ❞❡ ❡q✉❛çõ❡s s✐♠✉❧tâ♥❡❛s✳ ◆❛s ✜❣✉r❛s ✷ ❡ ✸ ♣♦❞❡♠♦s ✈✐s✉❛❧✐③❛r ✉♠❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠ 3×3❣❡♥❡r✐❝❛♠❡♥t❡ ♠♦str❛❞❛ ♣♦r ▲❡✐❜♥✐③ ❡ ♦❜s❡r✈❛♠♦s q✉❡ ❡❧❛ s❡ ❛♣r♦①✐♠❛ ♠✉✐t♦ ❞❛ ❧✐♥❣✉❛❣❡♠ ✉t✐❧✐③❛❞❛ ❤♦❥❡ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✳
11 12 13
21 22 23
31 32 33
F ig2:M atriz representada por Leibniz
a11 a12 a13 a21 a22 a23 a31 a32 a33
F ig3: M atriz representada atualmente
❙❡❣✉♥❞♦ ❇♦②❡r ✭❬✸❪✱ ♣♣✳ ✷✾✼✲✷✾✽✮✱ ❡♠ ✶✼✷✾✱ ♦ ♠❛t❡♠át✐❝♦ ❡s❝♦❝ês ❈♦❧✐♥ ▼❛❝❧❛✉r✐♥ ✭✶✻✾✽✲✶✼✹✻✮ ❢♦✐ ♦ ♣r✐♠❡✐r♦ ❛ ❣❡♥❡r❛❧✐③❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡ ♥❛ s♦❧✉çã♦ ❞❡ s✐st❡✲ ♠❛s ❧✐♥❡❛r❡s ❞❡n ❡q✉❛çõ❡s ❝♦♠ n ✐♥❝ó❣♥✐t❛s✱ ♣♦ré♠ s❡✉s ❡st✉❞♦s só ❢♦r❛♠ ♣✉❜❧✐❝❛❞♦s
❡♠ ✶✼✹✽✱ ❞♦✐s ❛♥♦s ❛♣ós s✉❛ ♠♦rt❡✱ ♥♦ tr❛❜❛❧❤♦ ✐♥t✐t✉❧❛❞♦ ✏❚r❡❛t✐s❡ ♦❢ ❛❧❣❡❜r❛✑✳ ▼❡s♠♦ t❡♥❞♦ s✐❞♦ ❡st✉❞❛❞❛ ♣♦r ▼❛❝❧❛✉r✐♥✱ ♦ ❛❧❣♦r✐t♠♦ ❞❡ r❡s♦❧✉çã♦ só ❣❛♥❤♦✉ r❡❛❧♠❡♥t❡ ♥♦t♦✲ r✐❡❞❛❞❡ ❛tr❛✈és ❞♦s ❡st✉❞♦s ❞♦ s✉íç♦ ●❛❜r✐❡❧ ❈r❛♠❡r ✭✶✼✵✹✲✶✼✺✷✮✳ ❖ ♠❛t❡♠át✐❝♦ s✉íç♦ ❝❤❡❣♦✉ ❛ ♣r❛t✐❝❛♠❡♥t❡ ♦s ♠❡s♠♦ r❡s✉❧t❛❞♦s ❞❡ ▼❛❝❧❛✉r✐♥ ❡♠ s✉❛ ♦❜r❛ ✏■♥tr♦❞✉çã♦ ❡ ❆♥á❧✐s❡ ❞❛s ❝✉r✈❛s ❆❧❣é❜r✐❝❛s✑ ❞❛t❛❞❛ ❞❡ ✶✼✺✵✱ ♣♦ré♠ ❝❛❜❡ s❛❧✐❡♥t❛r q✉❡ ♦ ♦❜❥❡t✐✈♦ ❞♦s ❡st✉❞♦s ❞❡ ❈r❛♠❡r ❡r❛ ❢♦❝❛❞♦ ❡♠ ❞❡t❡r♠✐♥❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ❝ô♥✐❝❛s ❡ s❡✉ ❞❡s❡♥✈♦❧✲ ✈✐♠❡♥t♦ ❞♦ ❡st✉❞♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ❢♦✐ ❛♣❡♥❛s ✉♠ ♠❡✐♦ ♣❛r❛ ❛t✐♥❣✐r ❡ss❡ ♦❜❥❡t✐✈♦✳ ❉❡ q✉❛❧q✉❡r ❥❡✐t♦ ❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ❞❡n❡q✉❛çõ❡s ❡n✐♥❝ó❣♥✐t❛s ♣♦r ♠❡✐♦ ❞♦
✉s♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s é ❛té ❤♦❥❡ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ✏❘❡❣r❛ ❞❡ ❈r❛♠❡r✑✳ ❆❝r❡❞✐t❛✲s❡ q✉❡ ❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ❞❡n ❡q✉❛çõ❡s ❡ n ✐♥❝ó❣♥✐t❛s ♣❡❧♦ ♠ét♦❞♦ ❞❡ ❈r❛♠♠❡r
t❡♥❤❛ ❣❛♥❤❛❞♦ ♠❛✐s ♥♦t♦r✐❡❞❛❞❡ q✉❡ ❛ ❛♣r❡s❡♥t❛❞❛ ♣♦r ▼❛❝❧❛✉r✐♥ ❞❡✈✐❞♦ ❛ ❞✐✈❡rs♦s
♠♦t✐✈♦s✱ ❡♥tr❡ ❡❧❡s ❛ s✉♣❡r✐♦r✐❞❛❞❡ ❞❛ ♥♦t❛çã♦ ✉t✐❧✐③❛❞❛ ♣♦r ❈r❛♠♠❡r ❡♠ ❞❡tr✐♠❡♥t♦ ❛ ❞❡ ▼❛❝❧❛✉r✐♥ ❡ t❛♠❜é♠ ❛♦ ❞❡❝❧í♥✐♦ ♠♦♠❡♥tâ♥❡♦ q✉❡ ♣❛ss❛✈❛ ❛ ❝♦♠✉♥✐❞❛❞❡ ❝✐❡♥tí✜❝❛ ❞❛ ■♥❣❧❛t❡rr❛ ♣❡r❛♥t❡ ❛ ❝♦♠✉♥✐❞❛❞❡ ❊✉r♦♣é✐❛✳
❙❡❣✉♥❞♦ ❇♦②❡r ✭❬✸❪✱ ♣♣✳ ✸✷✹✲✸✷✺✮✱ ❏♦s❡♣❤ ▲♦✉✐s ▲❛❣r❛♥❣❡ ✭✶✼✸✻✲✶✽✶✸✮ ❢♦✐ ♦ ♣r✐♠❡✐r♦ ❛ ✉t✐❧✐③❛r ♦ ❡st✉❞♦ ❞♦ ❞❡t❡r♠✐♥❛♥t❡ ❡♠ ✉♠❛ ❛♣❧✐❝❛çã♦ ❣❡♦♠étr✐❝❛ ❞✐r❡t❛✱ q✉❛♥❞♦ ❡♠ ✶✼✼✺✱ ♣✉❜❧✐❝♦✉ s❡✉ ❛rt✐❣♦ ✐♥t✐t✉❧❛❞♦ ✏❙♦❧✉t✐♦♥s ❛♥❛❧②t✐q✉❡s ❞❡ q✉❡❧q✉❡s s✉r ❧❡s ♣②r❛♠✐✲ ❞❡s tr✐❛♥❣✉❧❛✐r❡s✑ ♦♥❞❡ ❛♣❧✐❝❛✈❛ ♦ ❡st✉❞♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ♣❛r❛ ❣❡♥❡r❛❧✐③❛r ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r ❡ ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ t❡tr❛❡❞r♦ q✉❛❧q✉❡r✳ ❆s ✐❞❡✐❛s ❞❡ ▲❛❣r❛♥❣❡ ❛❜r✐r❛♠ ✉♠ ♣r❡❝❡❞❡♥t❡ ❣✐❣❛♥t❡s❝♦ ♣♦✐s ❛ ♣❛rt✐r ❞❛✐ ❝♦♠❡ç♦✉✲s❡ ❛ ♣❡r❝❡❜❡r ❝♦♠♦ ❛ á❧❣❡❜r❛ ❛♣❧✐❝❛❞❛ ♣♦❞❡r✐❛ t❡r ✈❛❧♦r s✐❣♥✐✜❝❛t✐✈♦ ♣❛r❛ ❛♣❧✐❝❛çõ❡s ❣❡♦♠étr✐❝❛s✳ P❡r❝❡❜❡♠♦s ❡ss❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ❢♦r♠❛ ✈❡❧❛❞❛ ❡♠ ✉♠❛ ❡①♣r❡ssã♦ ❞♦ ♣ró♣r✐♦ ▲❛❣r❛♥❣❡ q✉❛♥❞♦ ❡s❝r❡✈❡✉✱ ✏♣❛r❡❝❡✲♠❡ q✉❡ ❛s s♦❧✉çõ❡s q✉❡ ✈♦✉ ❛♣r❡s❡♥t❛r s❡rã♦ ❞❡ ✐♥t❡r❡ss❡ ♣❛r❛ ♦s ❣❡ô♠❡tr❛s t❛♥t♦ ♣❡❧♦s ♠ét♦❞♦s q✉❛♥t♦ ♣❡❧♦s r❡s✉❧t❛❞♦s✑✳
❙❡❣✉♥❞♦ ■❡③③✐ ❡ ❍❛③③❛♥ ✭❬✶❪✱ ♣♣✳ ✶✷✺✲✶✷✻✮✱ ❡♠ ✶✼✻✹✱ ♦ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês ➱t✐❡♥♥❡ ❇é③♦✉t ✭✶✼✸✵✲✶✼✽✸✮ ❝r✐♦✉ ✉♠ ♣r♦❝❡ss♦ ❞❡ ❡st❛❜❡❧❡❝✐♠❡♥t♦ ❞♦s s✐♥❛✐s ❞♦s t❡r♠♦s ❞❡ ✉♠ ❞❡t❡r♠✐♥❛♥t❡✱ ♠❛s ❢♦✐ s♦♠❡♥t❡ ❡♠ ✶✼✼✶ q✉❡ ❆❧❡①❛♥❞r❡ ❚❤❡ó♣❤✐❧❡ ❱❛♥❞❡r♠♦♥❞❡ ✭✶✼✸✺✲ ✶✼✾✻✮✱ t❛♠❜é♠ ❢r❛♥❝ês✱ ❡st✉❞♦✉ ❞❡ ❢♦r♠❛ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ❛ t❡♦r✐❛ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s✱ ❡st❡s ❢♦r❛♠ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❡st✉❞❛❞♦s ❝♦♠ ♠❛✐♦r ✐♠♣♦rtâ♥❝✐❛ ❡♠ s✐ ♠❡s♠♦s ❡ ♥ã♦ ❛♣❡♥❛s ❝♦♠♦ ♠❡✐♦ ♣❛r❛ ♦✉tr♦s ❡st✉❞♦s✳ ❙❡❣✉♥❞♦ ❇♦②❡r ✭❬✸❪✱ ♣♣✳ ✸✷✵✲✸✷✶✮✱ ❡♠ ✶✼✼✾✱ ❡♠ s❡✉ tr❛❜❛❧❤♦ ✐♥t✐t✉❧❛❞♦ ✏❚❤❡♦r✐❡ ❣❡♥❡r❛❧❡ ❞❡s ❡q✉❛t✐♦♥s ❛❧❣é❜r✐q✉❡s✑✱ ❇❡③♦✉t ✜❝♦✉ ❛✐♥❞❛ ♠❛✐s ❝♦♥❤❡❝✐❞♦ ♣♦r ❡st✉❞❛r ♦ ❞❡t❡r♠✐♥❛♥t❡ ♣❛r❛ ✉♠ tó♣✐❝♦ ❞❡♥♦♠✐♥❛❞♦ ❡❧✐♠✐♥❛çã♦ ❛❧❣é❜r✐❝❛✱ ❛ ♣❛rt✐r ❞❡ss❡ ❡st✉❞♦ ❡❧❡ ❞❡✉ r❡❣r❛s ❛rt✐✜❝✐❛✐s s❡♠❡❧❤❛♥t❡s ❛s ❞❡ ❈r❛♠♠❡r ♣❛r❛ r❡s♦❧✈❡r ✉♠ s✐st❡♠❛ ❞❡ n ❡q✉❛çõ❡s s✐♠✉❧tâ♥❡❛s ❡ n
✐♥❝ó❣♥✐t❛s ❡♠ q✉❡ s❡ ❞❡s❡❥❛ ❛❝❤❛r ❛ ❝♦♥❞✐çã♦ ❡♥tr❡ ♦s ❝♦❡✜❝✐❡♥t❡s ♥❡❝❡ssár✐❛ ♣❛r❛ q✉❡ ❛s ❡q✉❛çõ❡s ♣♦ss✉❛♠ ✉♠❛ s♦❧✉çã♦ ❝♦♠✉♠✳
❙❡❣✉♥❞♦ ■❡③③✐ ❡ ❍❛③③❛♥ ✭❬✶❪✱ ♣♣✳ ✶✷✺✲✶✷✻✮✱ ❡♠ ✶✼✼✷✱ P✐❡rr❡ ❙✐♠♦♥ ▲❛♣❧❛❝❡ ✭✶✼✹✾✲ ✶✽✷✼✮✱ ❡♠ s❡✉ ❛rt✐❣♦ ✏P❡sq✉✐s❛s s♦❜r❡ ♦ ❝á❧❝✉❧♦ ✐♥t❡❣r❛❧ ❡ ♦ s✐st❡♠❛ ❞♦ ♠✉♥❞♦✑ s✐♥t❡t✐③♦✉ ✉♠ ♠ét♦❞♦ q✉❡ ♣❡r♠✐t❡ ♦ ❝á❧❝✉❧♦ ❞♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ q✉❛❧q✉❡r ♠❛tr✐③ q✉❛❞r❛❞❛ ❛tr❛✈és ❞❡ ❞❡t❡r♠✐♥❛♥t❡s ♠❡♥♦r❡s ❞❡ ✉♠❛ ✜❧❛ q✉❛❧q✉❡r ❡s❝♦❧❤✐❞❛ ♥❛ ♠❛tr✐③ ❡ s❡✉s ❝♦♠♣❧❡♠❡♥✲ t♦s ❛❧❣é❜r✐❝♦s✳ ❆té ❤♦❥❡ ❡ss❡ ❚❡♦r❡♠❛ é ❣❧♦❜❛❧♠❡♥t❡ ❝♦♥❤❡❝✐❞♦ ❡ ✉t✐❧✐③❛❞♦ ❡♠ ❝á❧❝✉❧♦s ❞❡ ❞❡t❡r♠✐♥❛♥t❡s ❝♦♠♦ ❚❡♦r❡♠❛ ❞❡ ▲❛♣❧❛❝❡ ❡ ♣♦ss✉✐ ✐♠♣♦rtâ♥❝✐❛ tã♦ s✐❣♥✐✜❝❛t✐✈❛ q✉❡ ❛❧❣✉♥s ❛✉t♦r❡s ❞✐③❡♠ s❡r ♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❚❡♦r❡♠❛ ♥♦ ❡st✉❞♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s✳
❙❡❣✉♥❞♦ ❇♦②❡r ✭❬✸❪✱ ♣♣✳ ✸✺✸✲✸✺✾✮✱ ❡♠ ✶✽✶✷✱ ❆✉❣✉st✐♥✲▲♦✉✐s ❈❛✉❝❤② ✭✶✼✽✾✲✶✽✺✼✮ r❡✉♥✐✉ t✉❞♦ q✉❡ ❢♦r❛ ❢❡✐t♦ ❛té ♦ ♠♦♠❡♥t♦ s♦❜r❡ ♦ ❛ss✉♥t♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s ❡♠ ✉♠ ú♥✐❝♦ ❛rt✐❣♦ ❞❡ ♦✐t❡♥t❛ ❡ q✉❛tr♦ ♣á❣✐♥❛s ❛♣r❡s❡♥t❛❞♦ ❛ ❆❝❛❞❡♠✐❛ ❞❡ ❈✐ê♥❝✐❛s✱ ❛❧é♠ ❞❡ t❡r
s✐♠♣❧✐✜❝❛❞♦ ❡ ♠❡❧❤♦r❛❞♦ ✈❛r✐❛s ♥♦t❛çõ❡s✱ ❈❛✉❝❤② ❢♦✐ ♦ ♣r✐♠❡✐r♦ ♠❛t❡♠át✐❝♦ ❛ ❛tr✐❜✉✐r ❛ ♥♦♠❡♥❝❧❛t✉r❛ ✏❉❡t❡r♠✐♥❛♥t❡✑ ❛♦ ❡st✉❞♦ ❞❛ ❛❧❣♦r✐t✐♠✐③❛çã♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ n
❡q✉❛çõ❡s ❝♦♠n✐♥❝ó❣♥✐t❛s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❣❡♥ér✐❝❛ ❞❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r✳ ◆❛s❝✐❛ ❡♥tã♦
♦ ♥♦♠❡ ✏❉❊❚❊❘▼■◆❆◆❚❊✑ q✉❡ é ♦ ♥ú❝❧❡♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ss❡ tr❛❜❛❧❤♦✳ ❊♠❜♦r❛ ❢♦ss❡ ♥✉♠ ❝♦♥t❡①t♦ ❝♦♠♣❧❡t❛♠❡♥t❡ ❞✐❢❡r❡♥t❡✱ ❈❛✉❝❤② ❞❡r✐✈♦✉ ❛ ♥♦♠❡♥❝❧❛t✉r❛ ✏❞❡t❡r♠✐♥❛♥t❡✑ ❞❡ ✉♠ tr❛❜❛❧❤♦ ❞❡ ●❛✉ss✳ ❈❛✉❝❤② t❛♠❜é♠ ❞❡s❝r❡✈❡✉ ♦ ❡st✉❞♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❝♦♠♦ ✉♠❛ ❝❧❛ss❡ ❞❡ ❢✉♥çõ❡s s✐♠étr✐❝❛s ❛❧t❡r♥❛❞❛s ♦ q✉❡ ♥ã♦ tr❛❞✉③ ❜❡♠ ❛s r❡❢❡rê♥❝✐❛s ♠♦❞❡r♥❛s ❞♦ ❛ss✉♥t♦✳ ❊♠ ✉♠ ❛rt✐❣♦ ❞❛t❛❞♦ ❞❡ ✶✽✶✺ s♦❜r❡ ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s✱ ❈❛✉❝❤② ❡①tr❛♣♦❧♦✉ ♦ ✉s♦ ❡s♣❡❝í✜❝♦ ♥❛ á❧❣❡❜r❛ ❡ ❛♣❧✐❝♦✉ ♦ ❡st✉❞♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ❛ ❞♦✐s ♣r♦❜❧❡♠❛s✱ ✉♠ ❞❡ ❣❡♦♠❡tr✐❛ ❡ ✉♠ ❞❡ ❢ís✐❝❛ ♦♥❞❡ ✉t✐❧✐③♦✉ ♦ ❞❡t❡r♠✐♥❛♥t❡ ♣❛r❛ ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❡ t❛♠❜é♠ ❛ ❛❧❣✉♠❛s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s ❛ss♦❝✐❛❞❛s ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s✳ ❈❛✉❝❤② ❛✐♥❞❛ ✐♥tr♦❞✉③✐✉ ❛ ♥♦çã♦ s♦❜r❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s ❡ s✉❛s ❝♦♥s❡q✉ê♥❝✐❛s✳
❙❡❣✉♥❞♦ ❇♦②❡r ✭❬✸❪✱ ♣♣✳ ✸✻✶✲✸✻✹✮✱ ❛♣ós ❈❛✉❝❤②✱ ❈❛r❧ ●✉st❛✈ ❏❛❝♦❜✐ ✭✶✽✵✹✲✶✽✺✶✮ ❢♦✐ ♦ ♠❛t❡♠át✐❝♦ q✉❡ ♠❛✐s ❝♦♥tr✐❜✉✐✉ ♣❛r❛ ♦ ❛♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞♦ ❡st✉❞♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ❡ s✉❛s ♥♦t❛çõ❡s✳ P♦❞❡♠♦s ❛✜r♠❛r q✉❡ ❣r❛♥❞❡ ♣❛rt❡ ❞❛ ♥♦t❛çã♦ ♠♦❞❡r♥❛ q✉❡ ✉t✐❧✐③❛♠♦s ❤♦❥❡ ♣❛r❛ tr❛t❛r ♦ ❛ss✉♥t♦ ❢♦✐ ❛tr✐❜✉í❞❛ ♣♦r ❏❛❝♦❜✐ q✉❡ ❡r❛ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❣r❛♥❞❡ ❛❧❣♦r✐st❛ ❞❛ é♣♦❝❛✳ ❏❛❝♦❜✐ ❞❡✉ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ t❛♠❜é♠ ❛ ✉♠ ❡st✉❞♦ ❡s♣❡❝í✜❝♦ ❞❡♥tr♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ❞❡♥♦♠✐♥❛❞♦ ❝♦♠♦ ❞❡t❡r♠✐♥❛♥t❡s ❢✉♥❝✐♦♥❛✐s✳ ❊♠ ✶✽✹✶✱ ❡❧❡ ♣✉❜❧✐❝♦✉ ✉♠ ❧♦♥❣♦ ❛rt✐❣♦ ✏❉❡ ❞❡t❡r♠✐♥❛♥t✐❜✉s ❢✉♥❝t✐♦♥❛❧✐❜✉s✑ ❡s♣❡❝✐✜❝❛♠❡♥t❡ ❞❡❞✐❝❛❞♦ ❛♦ ❞❡t❡r♠✐♥❛♥t❡ ❏❛❝♦❜✐❛♥♦ ❡ ❡♠ s✉❛s ❞❡♠♦♥str❛çõ❡s ♠♦str♦✉ ✉♠❛ ❢♦r♠❛ ♣rát✐❝❛ ❞❡ ✉t✐❧✐③❛r ♦s ❞❡t❡r♠✐♥❛♥t❡s ❢✉♥❝✐♦♥❛✐s ♣❛r❛ ❞❡t❡r♠✐♥❛r s❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡q✉❛çõ❡s é ♦✉ ♥ã♦ ✐♥❞❡♣❡♥❞❡♥t❡✳
❈❛♣ít✉❧♦ ✷
❘❡❝♦♥str✉✐♥❞♦ ♦ ❉❡t❡r♠✐♥❛♥t❡
❊st❡ ❝❛♣ít✉❧♦ ❝♦♥st✐t✉✐ ✉♠❛ ♣r♦♣♦st❛ ❞✐❞át✐❝❛ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❝♦♠ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❞✐❢❡r❡♥t❡ ❞❛ tr❛❞✐❝✐♦♥❛❧♠❡♥t❡ ✉t✐❧✐③❛❞❛✳
■r❡♠♦s ♦❜t❡r✱ ❛♣ós ✉♠❛ ♠❛♥✐♣✉❧❛çã♦ ❛❧❣é❜r✐❝❛ ❞❡ ❡q✉❛çõ❡s✱ ❛ ❡①♣r❡ssã♦ q✉❡ tr❛❞✉③ ♦ s✐❣♥✐✜❝❛❞♦ ❞♦ ❞❡t❡r♠✐♥❛♥t❡ ♣❡❧❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s✱ ♦✉ s❡❥❛✱ ♣❛r❛ ♦s ❝❛s♦s ❞❡ ♠❛tr✐③❡s ❞❡ ✷❛ ❡ ❞❡ ✸❛ ♦r❞❡♠✱ ✐r❡♠♦s ❞❡❞✉③✐r ❛ ❡①♣r❡ssã♦ ❞♦ ❞❡t❡r♠✐♥❛♥t❡ ❛ss♦❝✐❛❞♦
❡ ♥ã♦ s✐♠♣❧❡s♠❡♥t❡ ❞❡✜♥✐✲❧❛ ❝♦♠♦ ❣❡r❛❧♠❡♥t❡ é ♣r♦♣♦st♦ ♥♦ s✐st❡♠❛ tr❛❞✐❝✐♦♥❛❧ ❞❡ ❡♥s✐♥♦ q✉❡ ♣✉❞❡♠♦s ❝♦♥st❛t❛r ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❬✹❪✱ ❬✺❪✱ ❬✻❪✱ ❬✼❪ ❡ ❬✽❪ ✉t✐❧✐③❛❞♦s ♥❛s ❡s❝♦❧❛s ♣ú❜❧✐❝❛s ❞♦ ❉❋✳
✷✳✶ ❉❡t❡r♠✐♥❛♥t❡ ❞❡ ♦r❞❡♠ ✷
❙❡♥❞♦ R ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ♣❛r❛ ♦❜t❡r ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛s ♠❛tr✐③❡s ❞❡
✷❛ ♦r❞❡♠✱ ❞❡✈❡♠♦s ❛❧❣♦r✐t♠✐③❛r ❛ r❡s♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❞✉❛s ❡q✉❛çõ❡s ❡ ❞✉❛s
✐♥❝ó❣♥✐t❛s s♦❜r❡R✱ ❞❡ ❢♦r♠❛ ❣❡♥❡r❛❧✐③❛❞❛ ❛ ✜♠ ❞❡ ❝❤❡❣❛r♠♦s ❛ ❡①♣r❡ssã♦ ❞❡t❡r♠✐♥❛♥t❡
❞❡ q✉❛❧q✉❡r ♠❛tr✐③ ❞❡ ♦r❞❡♠ ✷✳
❱❛♠♦s ❝♦♥s✐❞❡r❛r ♦ s❡❣✉✐♥t❡ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s✿
I :ax+by =f
II :cx+dy=g
♦♥❞❡ x ❡ y r❡♣r❡s❡♥t❛♠ ❛s ✐♥❝ó❣♥✐t❛s ❞♦ s✐st❡♠❛✱ a ❡ c r❡♣r❡s❡♥t❛♠ ♦s ❝♦❡✜❝✐❡♥t❡s
❞❡x✱b❡dr❡♣r❡s❡♥t❛♠ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡y❡f ❡gr❡♣r❡s❡♥t❛♠ ♦s t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s
❞♦ s✐st❡♠❛✱ t♦❞♦s ❡❧❡s ❡❧❡♠❡♥t♦s ❞❡ R✱ t❡♥t❛r❡♠♦s ❛❧❣♦r✐t♠✐③❛r ❛ r❡s♦❧✉çã♦ ♣♦r ♠❡✐♦
❞♦ ♣r♦❝❡ss♦ ❞❡ ❡❧✐♠✐♥❛çã♦ ❞❡ ✐♥❝ó❣♥✐t❛s✳
❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ✈❛♠♦s ❡❧✐♠✐♥❛r ❛ ✐♥❝ó❣♥✐t❛y❡ ♣❛r❛ ✐ss♦ ♠✉❧t✐♣❧✐❝❛r❡♠♦s
t♦❞❛ ❛ ❡q✉❛çã♦II ♣♦r −b
d ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ r❡❛❧✐③❛r♠♦s ❡ss❛ ♦♣❡r❛çã♦ ❞❡✈❡♠♦s t❡rd6= 0✳
P♦ré♠ s❡d= 0✱x t❡rá ✉♠ ✈❛❧♦r ❜❡♠ ❞❡✜♥✐❞♦✱ ❛ s❛❜❡r✱x= gc ❡ ❞❡ss❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s s✉❜st✐t✉✐r ♦ ✈❛❧♦r ❞❡x♥❛ s❡❣✉♥❞❛ ❡q✉❛çã♦ ❡ ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞❡y✳ ❈❛❜❡ s❛❧✐❡♥t❛r q✉❡c
❡d♥ã♦ ♣♦❞❡♠ s❡r ❛♦ ♠❡s♠♦ t❡♠♣♦ ✐❣✉❛✐s ❛ ③❡r♦✱ ♣♦✐s s❡ ❢♦r❡♠✱ t❡r❡♠♦s ✉♠❛ ❡q✉❛çã♦
❞♦ t✐♣♦0x+ 0y=g✳ ❙❡g = 0♦ s✐st❡♠❛ t❡rá ❛♣❡♥❛s ✉♠❛ ❡q✉❛çã♦✱ ♦ q✉❡ ♥ã♦ é ♦ ♦❜❥❡t♦ ❞♦ ❡st✉❞♦❀ ❡ s❡ g 6= 0 ♦ s✐st❡♠❛ ♥ã♦ t❡rá s♦❧✉çã♦✳
❙✉♣♦♥❞♦ ❡♥tã♦ d6= 0✱ ❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ ■■ ♣♦r −b
d✱ ♦❜t❡♠♦s ❛ ❡q✉❛çã♦
IIb :
−bc
d x−by=
−bg d .
❙♦♠❛♥❞♦ ❛❣♦r❛ ❛ ❡q✉❛çã♦I ❝♦♠ ❛ ❡q✉❛çã♦ IIb✱ ❛ ✐♥❝ó❣♥✐t❛ y é ❡❧✐♠✐♥❛❞❛ ❡ ❛ ❡q✉❛çã♦
r❡s✉❧t❛♥t❡ ♣♦❞❡ s❡r ❞❡s❡♥✈♦❧✈✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
x(a−bc
d) = f− bg
d ⇒x(
ad−bc d ) =
f d−bg
d ⇒x(ad−bc) =f d−bg.
❈♦♠ ✐ss♦✱ ♣♦❞❡♠♦s ❡①♣r❡ss❛r ❣❡♥❡r✐❝❛♠❡♥t❡ ♦ ✈❛❧♦r ❞❛ ✐♥❝ó❣♥✐t❛x❛tr❛✈és ❞❛ ❝♦♠✲
❜✐♥❛çã♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❡ ❞♦s t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❞♦ s✐st❡♠❛ ♣♦r ♠❡✐♦ ❞❛ ❡①♣r❡ssã♦
x(ad−bc) = (f d−bg)
❆♥❛❧✐s❛♥❞♦ ❛ ❡①♣r❡ssã♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ x só t❡rá s♦❧✉çã♦ ❝❛s♦ ad−bc 6= 0✳ ❊ss❛ ❛♥á❧✐s❡ ♠♦str❛ ♦ q✉❡ ❥á ❤❛✈✐❛♠♦s ♠❡♥❝✐♦♥❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡ s♦❜r❡ ❛ ✐♠♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡c❡ d s❡r❡♠ ❛♦ ♠❡s♠♦ t❡♠♣♦ ♥✉❧♦s✳
❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♠❛♥✐♣✉❧❛♠♦s ❛s ❡q✉❛çõ❡s ♣❛r❛ ❡❧✐♠✐♥❛r ❛ ✐♥❝ó❣♥✐t❛ y ❡ ❡♥✲
❝♦♥tr❛r ❛ s♦❧✉çã♦ ♣❛r❛ x✱ ♣♦❞❡♠♦s r❡♣❡t✐r ♦ ♣r♦❝❡ss♦✱ ♣♦ré♠ ❞❡ss❛ ✈❡③ ♠❛♥✐♣✉❧❛♠♦s
❛s ❡q✉❛çõ❡s ❛ ✜♠ ❞❡ ❡❧✐♠✐♥❛r♠♦s ❛ ✐♥❝ó❣♥✐t❛x ❡ ❝♦♠ ✐ss♦ ❡♥❝♦♥tr❛r ❛ s♦❧✉çã♦ ♣❛r❛y✳
P❛r❛ ✐ss♦ ✐♥✐❝✐❛♠♦s ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❊q✉❛çã♦ II ♣♦r −a
c ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ r❡❛❧✐③❛r♠♦s
❡ss❛ ♦♣❡r❛çã♦ ❞❡✈❡♠♦s s✉♣♦r q✉❡ c 6= 0✱ ❛♥❛❧♦❣❛♠❡♥t❡ ❛♦ q✉❡ ✜③❡♠♦s ♥❛ ♣r✐♠❡✐r❛ ♠❛♥✐♣✉❧❛çã♦ ❛❧❣é❜r✐❝❛✳
❙✉♣♦♥❞♦ ❡♥tã♦ c6= 0✱ ❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ ■■ ♣♦r −a
c ✱ ♦❜t❡♠♦s ❛ ❡q✉❛çã♦
IIc :−ax− ad
c y=
−ag c .
❙♦♠❛♥❞♦ ❛❣♦r❛ ❛ ❡q✉❛çã♦I ❝♦♠ ❛ ❡q✉❛çã♦ IIc✱ ❛ ✐♥❝ó❣♥✐t❛ ① é ❡❧✐♠✐♥❛❞❛ ❡ ❛ ❡q✉❛çã♦
r❡s✉❧t❛♥t❡ ♣♦❞❡ s❡r ❞❡s❡♥✈♦❧✈✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
y(b− ad
c ) = f− ag
c ⇒y(
bc−ad c ) =
f c−ag
c ⇒y(bc−ad) = f c−ag.
❈♦♠ ✐ss♦✱ ♣♦❞❡♠♦s ❡①♣r❡ss❛r ❣❡♥❡r✐❝❛♠❡♥t❡ ♦ ✈❛❧♦r ❞❛ ✐♥❝ó❣♥✐t❛y❛tr❛✈és ❞❛ ❝♦♠✲
❜✐♥❛çã♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❡ ❞♦s t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❞♦ s✐st❡♠❛ ♣♦r ♠❡✐♦ ❞❛ ❡①♣r❡ssã♦
y(ad−bc) = (ag−f c)
❆♥❛❧♦❣❛♠❡♥t❡ ❛♦ ❡st✉❞♦ q✉❡ ✜③❡♠♦s ♣❛r❛ ❛ ❡q✉❛çã♦ q✉❡ ❞❡t❡r♠✐♥❛ ❛ ✐♥❝ó❣♥✐t❛ x✱
♣❡r❝❡❜❡♠♦s q✉❡y só t❡rá s♦❧✉çã♦ ❝❛s♦ad−bc6= 0✳
❆ ♣❛rt✐r ❞✐ss♦✱ ❡ s✉♣♦♥❞♦ ad−bc6= 0✱ ♣♦❞❡♠♦s ❣❡♥❡r❛❧✐③❛r ❛ r❡s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ♥❛s ✐♥❝ó❣♥✐t❛sx ❡ y ❝♦♠♦ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❡ ❝♦❡✜❝✐❡♥t❡s ♦♥❞❡
x(ad−bc) = (f d−bg) ; y(ad−bc) = (ag−f c)
❖❜s❡r✈❡ q✉❡ ❛ ❡①♣r❡ssã♦ q✉❡ ❢❛③ ♦ ♣❛♣❡❧ ❞❡ ❝♦❡✜❝✐❡♥t❡ ❞❡x❡ ❞❡y♥❛s ❞✉❛s ❡q✉❛çõ❡s
é ❛ ♠❡s♠❛✱ ❡ ♣♦r ❡ss❡ ♠♦t✐✈♦ ❝❤❛♠❛r❡♠♦s ❡st❛ ❡①♣r❡ssã♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ♦r❞❡♠ ✷✳ ❯s❛r❡♠♦s ❛❧❣✉♠❛s ✈❡③❡s ❛ ❡①♣r❡ssã♦ Det ♣❛r❛ ♥♦s r❡❢❡r✐r♠♦s ❛♦ ✈❛❧♦r ❞❡t❡r♠✐♥❛♥t❡✳
❉❡ss❛ ❢♦r♠❛ t❡♠♦s
❙❡ ❛❣r✉♣❛r♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ✐♥❝ó❣♥✐t❛sx❡y r❡s♣❡❝t✐✈❛♠❡♥t❡ ❡♠ ✉♠❛ ♠❛tr✐③✱
♦❜s❡r✈❛r❡♠♦s q✉❡ ♦ Det é ❞❛❞♦ ♣❡❧❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ♣r♦❞✉t♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s q✉❡
❡stã♦ ❛❧♦❝❛❞♦s ♥❛s ❞✉❛s ❞✐❛❣♦♥❛✐s ❞❛ ♠❛tr✐③✳ ▼❛✐s ❛ ❢r❡♥t❡✱ tr❛t❛r❡♠♦s ❡ss❛s ❞✐❛❣♦♥❛✐s ♣♦r ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧ ❡ ❞✐❛❣♦♥❛❧ s❡❝✉♥❞ár✐❛ ❡ ❞❡✜♥✐r❡♠♦s s❡✉s ❡❧❡♠❡♥t♦s ❞❡ ❢♦r♠❛ ✐♥❞❡①❛❞❛✳
❯t✐❧✐③❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❡ ♥♦t❛çã♦ ❞❡ ♠❛tr✐③ ❡ ❞❡ ❞❡t❡r♠✐♥❛♥t❡ ❛❜♦r❞❛❞❛ ♥♦ ❝❛♣✐✲ t✉❧♦ ✸✱ ✈❛♠♦s ❛❣r✉♣❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❞♦ s✐st❡♠❛ ❧✐♥❡❛r ❛ ✜♠ ❞❡ ❢❛❝✐❧✐t❛r ♦ ❝á❧❝✉❧♦ ❞♦ ❞❡t❡r♠✐♥❛♥t❡✳
❆ss✐♠✱ ♥❛ ♠❛tr✐③
a b
c d
✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ss❛ ♠❛tr✐③ é
❞❛❞♦ ♣♦rDet=
a b
c d
=ad−bc✳
❖❜s❡r✈❛♠♦s ❛✐♥❞❛ q✉❡ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❞❛ ✐♥❝ó❣♥✐t❛ x✱ ♥ã♦ ❜❛st❛ ❛♣❡♥❛s
♦ ✈❛❧♦r ❞❡t❡r♠✐♥❛♥t❡ Det✱ ♣r❡❝✐s❛r❡♠♦s t❛♠❜é♠ ❞♦ ♥✉♠❡r❛❞♦r ❞❛ ❢r❛çã♦ q✉❡ é ❞❛❞♦
♣❡❧❛ ❡①♣r❡ssã♦f d−bg✳ ❆ ❡st❡ ✈❛❧♦r ❞❛r❡♠♦s ♦ ♥♦♠❡ ❞❡ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ x❡ ✉s❛r❡♠♦s
❛ ❡①♣r❡ssã♦
Detx=f d−bg.
❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ✉t✐❧✐③❛r❡♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s ♣❛r❛ ❢♦r♠❛r ✉♠❛ ♠❛tr✐③✱ ♣♦ré♠ tr♦❝❛r❡♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ✐♥❝ó❣♥✐t❛ x ♣❡❧♦s t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❞♦ s✐st❡♠❛✱ ❡
❛ss✐♠ ♣♦❞❡♠♦s ❡s❝r❡✈❡r q✉❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ x é ♦ Det ❞❛ ♠❛tr✐③ ♦❜t✐❞❛ ❝♦♠ ♦s
t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ y✳
❆ss✐♠✱ ♥❛ ♠❛tr✐③
f b
g d
✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ss❛ ♠❛tr✐③ é
❞❛❞♦ ♣♦rDetx =
f b
g d
=f d−bg✳
❈♦♠ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ♦❜s❡r✈❛çã♦✱ ♣♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❞❛ ✐♥❝ó❣♥✐t❛y✱ ♣r❡❝✐s❛r❡♠♦s ❞♦ ♥✉♠❡r❛❞♦r q✉❡ é ❞❛❞♦ ♣❡❧❛ ❡①♣r❡ssã♦ag−f c✳ ❆ ❡st❡
✈❛❧♦r ❞❛r❡♠♦s ♦ ♥♦♠❡ ❞❡ ❞❡t❡r♠✐♥❛♥t❡ ❞❡y ❡ ✉s❛r❡♠♦s ❛ ❡①♣r❡ssã♦
Dety =ag−f c.
❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ✉t✐❧✐③❛r❡♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s ♣❛r❛ ❢♦r♠❛r ✉♠❛ ♠❛tr✐③✱ ♣♦ré♠ tr♦❝❛r❡♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ✐♥❝ó❣♥✐t❛ y ♣❡❧♦s t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❞♦ s✐st❡♠❛✱ ❡
❛ss✐♠ ♣♦❞❡♠♦s ❡s❝r❡✈❡r q✉❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ y é ♦ Det ❞❛ ♠❛tr✐③ ♦❜t✐❞❛ ❝♦♠ ♦s
t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ y✳
❆ss✐♠✱ ♥❛ ♠❛tr✐③
a f
c g
✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ss❛ ♠❛tr✐③ é
❞❛❞♦ ♣♦rDety =
a c
f g
=ag−f c✳
❈❤❡❣❛♥❞♦ ❛ ❡ss❡ ♣♦♥t♦ ❞♦ ❡st✉❞♦ ♣♦❞❡♠♦s ❣❡♥❡r❛❧✐③❛r ❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❞❡ ❞✉❛s ❡q✉❛çõ❡s ❡ ❞✉❛s ✐♥❝ó❣♥✐t❛sx ❡ y ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❉❛❞♦ ♦ s✐st❡♠❛ ❧✐♥❡❛r ❞❡ ❞✉❛s ❡q✉❛çõ❡s ❡ ❞✉❛s ✐♥❝ó❣♥✐t❛s
ax+by =f
cx+dy =g
❛s s♦❧✉çõ❡s ❞♦ s✐st❡♠❛ ♣♦❞❡♠ s❡r ❣❡♥❡r✐❝❛♠❡♥t❡ ❡①♣r❡ss❛s ♣❡❧❛s r❡❧❛çõ❡s
x(Det) =Detx ; y(Det) =Dety
♦♥❞❡✿
Deté ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③
a b
c d
❢♦r♠❛❞❛ ♣❡❧♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ✐♥❝ó❣♥✐t❛s
x❡ y r❡s♣❡❝t✐✈❛♠❡♥t❡ ❡✱ Det=
a b
c d
=ad−bc❀ Detx é ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③
f b
g d
❢♦r♠❛❞❛ ♣❡❧♦s t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❡
♣❡❧♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ✐♥❝ó❣♥✐t❛y ❡✱ Detx =
f b
g d
=f d−bg❀ Dety é ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③
a f
c g
❢♦r♠❛❞❛ ♣❡❧♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ✐♥❝ó❣♥✐t❛
x❡ ♣❡❧♦s t❡r♠♦s ✐♥❞❡♣❡♥❞❡st❡ ❞♦ s✐st❡♠❛ ❡✱ Dety =
a f
c g
=ag−f c✳
❈❛❜❡ s❛❧✐❡♥t❛r q✉❡ ❝♦♠♦x❡ ysã♦ ❞❡t❡r♠✐♥❛❞♦s ♣♦r ✉♠❛ r❛③ã♦✱ ❡st❛ ♣♦❞❡ ❛ss✉♠✐r
❝❛s♦s ❞✐❢❡r❡♥t❡s ♦s q✉❛✐s ❛❜♦r❞❛r❡♠♦s ❛ s❡❣✉✐r✳
✷✳✶✳✶ ❈❛s♦s ❞♦ ❙✐st❡♠❛
✶♦ ❈❆❙❖ ✲ (Det6= 0)
❙❡♠♣r❡ q✉❡ ♦ Det6= 0✱ ❛ ❡①♣r❡ssã♦ q✉❡ ❞❡t❡r♠✐♥❛ ❛ ✐♥❝ó❣♥✐t❛x ♣♦❞❡rá s❡r s♦❧✉❝✐✲
♦♥❛❞❛✱ ❧♦❣♦ ♦ s✐st❡♠❛ t❡rá s♦❧✉çã♦ ♣♦ssí✈❡❧ ❡ ✉♥✐❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦ ❛❧❣♦rít✐♠♦ ❞❡ r❡s♦❧✉çã♦ ❛♥t❡r✐♦r♠❡♥t❡ ❞❡♠♦♥str❛❞♦✳
✷♦ ❈❆❙❖ ✲ (Det= 0, Det
x = 0 e Dety = 0)
❙❡♠♣r❡ q✉❡ t♦❞♦s ♦s ❞❡t❡r♠✐♥❛♥t❡s ❞♦ s✐st❡♠❛ ❢♦r❡♠ ♥✉❧♦s✱ ❛s ❡①♣r❡ssõ❡s q✉❡ ❞❡✲ t❡r♠✐♥❛♠x ❡y s❡rã♦ r❡♣r❡s❡♥t❛❞❛s ❝♦♠♦ 0x= 0 ❡ 0y= 0 ❡ t❛✐s ❡①♣r❡ssõ❡s ♣♦❞❡♠ t❡r ✐♥✜♥✐t❛s s♦❧✉çõ❡s✳ ▲♦❣♦ ♦ s✐st❡♠❛ s❡rá ✐♥❞❡t❡r♠✐♥❛❞♦ ❡ t❡rá ✐♥✜♥✐t♦s ♣❛r❡s ❞❡ s♦❧✉çã♦ (x, y)✳
✸♦ ❈❆❙❖ ✲ (Det= 0, Det
x 6= 0 ou Dety 6= 0)
❙❡♠♣r❡ q✉❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❢♦r ♥✉❧♦ ❡ q✉❡ ♦s ❞❡t❡r♠✐♥❛♥t❡s ❞❡x♦✉ ❞❡y❢♦r❡♠ ❞✐❢❡✲
r❡♥t❡s ❞❡ ③❡r♦✱ ❛s ❡①♣r❡ssõ❡s q✉❡ ❞❡t❡r♠✐♥❛♠ ❛s ✐♥❝ó❣♥✐t❛sx ♦✉y s❡rã♦ r❡♣r❡s❡♥t❛❞❛s
♣♦r0x=Detx ♦✉ 0y=Dety✳ ❚❛✐s ❡①♣r❡ssõ❡s ♥ã♦ ❛♣r❡s❡♥t❛♠ s♦❧✉çã♦✱ ❧♦❣♦ ♦ s✐st❡♠❛
s❡rá ✐♠♣♦ssí✈❡❧✳
✷✳✶✳✷ ■♥❞❡①❛♥❞♦ ♦s ❊❧❡♠❡♥t♦s ❞❛ ▼❛tr✐③
❆ ✜♠ ❞❡ ❞❡✜♥✐r ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ♠❛tr✐③✱ s✉r❣✐✉ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❣❡♥❡r❛❧✐③❛r s❡✉s ❡❧❡♠❡♥t♦s ❞❡ ❢♦r♠❛ ♠❛✐s ❝r✐t❡r✐♦s❛✳ P❛r❛ ✐ss♦ ❢♦✐ ❝r✐❛❞♦ ✉♠❛ ❢♦r♠❛ ❞❡ ✐♥❞❡①❛r ♦s ❡❧❡♠❡♥t♦s ❞❛ ♠❛tr✐③ ♦♥❞❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ r❡❝❡❜❡ ✉♠❛ ❧❡tr❛ r❡♣r❡s❡♥t❛t✐✈❛ ❞♦ ❡❧❡♠❡♥t♦ ❡ ✉♠ í♥❞✐❝❡ q✉❡ r❡♣r❡s❡♥t❛ ❛ ❧✐♥❤❛ ❡ ❛ ❝♦❧✉♥❛ ♦♥❞❡ ♦ ❡❧❡♠❡♥t♦ s❡ ❡♥❝♦♥tr❛ ♥❛ ♠❛tr✐③✳
❆ss✐♠✱ q✉❛♥❞♦ ✐♥❞❡①❛♠♦s ❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠ ✷ q✉❡ ❢♦✐ ✉t✐❧✐③❛❞❛ ♣❛r❛ r❡s♦❧✈❡r ♦ s✐st❡♠❛ ❞❡ ❞✉❛s ❡q✉❛çõ❡s ❡ ❞✉❛s ✐♥❝ó❣♥✐t❛s t❡♠♦s
a b
c d
=
a11 a12 a21 a22
P♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ aij ❞❛ ♠❛tr✐③ r❡♣r❡s❡♥t❛ ♦ ❡❧❡♠❡♥t♦ ❛❧♦❝❛❞♦
♥❛ ❧✐♥❤❛ i ❡ ♥❛ ❝♦❧✉♥❛ j✳ ❉❡ss❛ ❢♦r♠❛ ♣♦❞❡✲s❡ ❡st❛❜❡❧❡❝❡r ❞♦✐s ❝♦♥❝❡✐t♦s ✐♠♣♦rt❛♥t❡s
❞❡♥tr♦ ❞♦ ❡st✉❞♦ ❞❛s ♠❛tr✐③❡s✳
❖ ♣r✐♠❡✐r♦ ❡st❛❜❡❧❡❝❡ ❛ ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧ ❞❡ q✉❛❧q✉❡r ♠❛tr✐③ q✉❡ é ♦ ❝♦♥❥✉♥t♦ ❞❡ ❡❧❡♠❡♥t♦s aij ❞❡st❛ ♠❛tr✐③ ♦♥❞❡ i = j✳ ❆ss✐♠ ♥❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠ ✷✱ ❛ ❞✐❛❣♦♥❛❧
♣r✐♥❝✐♣❛❧ é ❞❛❞❛ ♣♦r
Dp ={a11, a22}
❖ s❡❣✉♥❞♦ ❡st❛❜❡❧❡❝❡ ❛ ❞✐❛❣♦♥❛❧ s❡❝✉♥❞ár✐❛ ❞❡ q✉❛❧q✉❡r ♠❛tr✐③ q✉❡ é ♦ ❝♦♥❥✉♥t♦ ❞❡ ❡❧❡♠❡♥t♦saij ❞❡st❛ ♠❛tr✐③ ♦♥❞❡i+j é ✐❣✉❛❧ ❛ ♦r❞❡♠ ❞❛ ♠❛tr✐③ ♠❛✐s ✉♠✳ ❆ss✐♠ ♥❛
♠❛tr✐③ ❞❡ ♦r❞❡♠ ✷✱ ❛ ❞✐❛❣♦♥❛❧ s❡❝✉♥❞ár✐❛ é ❞❛❞❛ ♣♦r
Ds={a12, a21}
❆♣ós ❡st❛❜❡❧❡❝✐❞♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ ❞✐❛❣♦♥❛✐s ❞❛ ♠❛tr✐③✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ♦ ❞❡✲ t❡r♠✐♥❛♥t❡ ❞❡ss❛ ♠❛tr✐③ ❝♦♠♦
Det=
a11 a12
a21 a22
=a11a22−a21a12
❖✉tr❛ ❢♦r♠❛ ❞❡ ❡s❝r❡✈❡r ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ss❛ ♠❛tr✐③ é ❞❡✜♥✐✲❧♦ ❝♦♠♦ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ♣r♦❞✉t♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❛ ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧ ❡ ♦ ♣r♦❞✉t♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❛ ❞✐❛❣♦♥❛❧ s❡❝✉♥❞ár✐❛✳
✷✳✶✳✸ ❆♥❛❧♦❣✐❛ ●❡♦♠étr✐❝❛ ❞♦s ❈❛s♦s ❞♦ ❙✐st❡♠❛
❈❛❞❛ ✉♠ ❞♦s ❝❛s♦s ❞♦ s✐st❡♠❛ ❞❡ ❞✉❛s ❡q✉❛çõ❡s ❡ ❞✉❛s ✐♥❝ó❣♥✐t❛s q✉❡ ❢♦r❛♠ ♠♦s✲ tr❛❞♦s ♥❡st❛ s✉❜s❡çã♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ♣♦r ♣♦s✐çõ❡s r❡❧❛t✐✈❛s ❡♥tr❡ ❞✉❛s r❡t❛s ♥♦ ♣❧❛♥♦✳ ❊st❛ ❛♥❛❧♦❣✐❛ ❛tr✐❜✉✐ s✐❣♥✐✜❝❛❞♦ ❛❞✐❝✐♦♥❛❧ ❛♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦s ❝❛s♦s ❞♦ s✐st❡♠❛ ❡ ♣♦❞❡ t❡r ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥♦ tr❛❜❛❧❤♦ ❝♦♠ ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❝❛s♦ s❡❥❛♠ ❛❜♦r❞❛❞❛s ❞❛ ❢♦r♠❛ ❡ ♥♦ ♠♦♠❡♥t♦ ❝♦rr❡t♦s✳
❈♦♠♦ ✈❡r✐✜❝❛♠♦s ♥❛s r❡❢❡rê♥❝✐❛s ❬✹❪✱ ❬✺❪✱ ❬✻❪✱ ❬✼❪ ❡ ❬✽❪✱ ❛ ♠❛✐♦r✐❛ ❞❛s ❡s❝♦❧❛s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛s ❞♦ ❡♥s✐♥♦ ♣ú❜❧✐❝♦✱ ♥ã♦ ❛ss♦❝✐❛♠ ❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❝♦♠ ❛s r❡t❛s ♥♦ ♣❧❛♥♦✱ tr❛t❛♥❞♦ ❡st❡s ❞♦✐s ❛ss✉♥t♦s ❞❡ ❢♦r♠❛ t♦t❛❧♠❡♥t❡ s❡♣❛r❛❞❛ ❡ s❡♠ ♥❡♥❤✉♠ t✐♣♦ ❞❡ ❛♥❛❧♦❣✐❛ ♣♦st❡r✐♦r✳ P❛r❡❝❡ ó❜✈✐♦✱ ♠❛s ♠✉✐t♦s ♥ã♦ ♣❡r❝❡❜❡♠ ♦ ❣r❛♥❞❡ ♣♦t❡♥❝✐❛❧ ❞❡ ❡♥s✐♥♦ ❡ ❝♦rr❡❧❛çã♦ ❞❡ ❝♦♥t❡ú❞♦s q✉❡ ♣♦ss✉✐ ❡st❡ ❛ss✉♥t♦✱ ♣♦❞❡♥❞♦ tr❛❜❛❧❤❛r ❝♦♥❥✉♥t❛♠❡♥t❡ ❞✐✈❡rs♦s tó♣✐❝♦s ❞♦ ❡♥s✐♥♦ ♠❛t❡♠át✐❝♦ ❝♦♠♦ ❡st✉❞♦ ❞❛s r❡t❛s ♥♦ ♣❧❛♥♦✱ ❢✉♥çõ❡s✱ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞♦ ♣r✐♠❡✐r♦ ❣r❛✉✱ ♠❛tr✐③❡s ❡ ❞❡t❡r♠✐♥❛♥t❡s✱ ❞❡♥tr❡ ♦✉tr♦s✳
◆♦s ♣❛rá❣r❛❢♦s ❛❜❛✐①♦✱ ❛❜♦r❞❛r❡♠♦s ❞❡ ❢♦r♠❛ s✉♣❡r✜❝✐❛❧ ✉♠❛ ❛♥❛❧♦❣✐❛ ❣❡♦♠étr✐❝❛ ❡♥tr❡ r❡t❛s ♥♦ ♣❧❛♥♦ ❛ ❝❛❞❛ ✉♠ ❞♦s ❝❛s♦s q✉❡ ♦ s✐st❡♠❛ ❞❡ ❞✉❛s ❡q✉❛çõ❡s ❡ ❞✉❛s ✐♥❝ó❣✲ ♥✐t❛s ♣♦❞❡ ❛ss✉♠✐r✱ s❛❧✐❡♥t❛♥❞♦ q✉❡ ❡ss❛ ❛❜♦r❞❛❣❡♠ ♣♦❞❡ s❡r ❜❡♠ ♠❛✐s ❡①♣❧♦r❛❞❛ ♣❡❧♦ ♣r♦❢❡ss♦r ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❝❛s♦ ❝♦♥st❛t❡ q✉❡ ❛ t✉r♠❛ t❡♠ ♦s ♣ré r❡q✉✐s✐t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ t❛❧✳
✶♦ ❈❆❙❖ ✲ (Det6= 0)
❈♦♠♦ ❥❛ ✈✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ♥❡st❡ ❝❛s♦ ❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ é ú♥✐❝❛ ❡ ❞❡t❡r♠✐✲ ♥❛❞❛✱ ♦✉ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ só ❡①✐st❡ ✉♠ ♣❛r ♦r❞❡♥❛❞♦(x, y)q✉❡ s♦❧✉❝✐♦♥❛ ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s✳
❆♦ ❢❛③❡r♠♦s ✉♠❛ ❛♥❛❧♦❣✐❛ ❝♦♠ ♦ ❡st✉❞♦ ❞❛s r❡t❛s✱ r❡♣r❡s❡♥t❛♠♦s ❝❛❞❛ ✉♠❛ ❞❛s ❡q✉❛çõ❡s ❞♦ s✐st❡♠❛ ❝♦♠♦ ✉♠❛ ❡q✉❛çã♦ ❞❛ r❡t❛ ♥♦ ♣❧❛♥♦✱ ❡ ❝♦♠ ✐ss♦ ❛s s♦❧✉çõ❡s ❞♦ s✐st❡♠❛ s❡rã♦ ♣❛r❡s ♦r❞❡♥❛❞♦s (x, y) q✉❡ ❡st❡❥❛♠ ❛♦ ♠❡s♠♦ t❡♠♣♦ ♥❛s ❞✉❛s r❡t❛s✳ ❈♦♠♦ ♥❡st❡ ❝❛s♦ ❛ s♦❧✉çã♦ é ú♥✐❝❛✱ só ♣♦❞❡ ❡①✐st✐r ✉♠ ♣♦♥t♦ ♥♦ ♣❧❛♥♦ q✉❡ ❡st❡❥❛ ❝♦♥t✐❞♦ ❛♦ ♠❡s♠♦ t❡♠♣♦ ♥❛s ❞✉❛s r❡t❛s✳
❊st✉❞❛♥❞♦ ❛s três ♣♦s✐çõ❡s r❡❧❛t✐✈❛s ♣♦ssí✈❡✐s ❡♥tr❡ ❞✉❛s r❡t❛s ♥♦ ♣❧❛♥♦✱ ♣❡r❝❡❜❡♠♦s q✉❡ ❛ ú♥✐❝❛ q✉❡ ❛t❡♥❞❡ ❛ ❡st❡ ❝❛s♦ ❡s♣❡❝í✜❝♦ é ❛ ❞❡ r❡t❛s ❝♦♥❝♦rr❡♥t❡s ❡♠ ✉♠ ♣♦♥t♦✱ ❡ ❡st❡ ♣♦♥t♦ s❡rá t❛♠❜é♠ ❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❝♦♠♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ♥❛ ✜❣✉r❛ ✷✳✶
❋✐❣✉r❛ ✷✳✶✿ r❡t❛s ❝♦♥❝♦rr❡♥t❡s
✷♦ ❈❆❙❖ ✲ (Det= 0, Det
x = 0 e Dety = 0)
❈♦♠♦ ❥❛ ✈✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ♥❡st❡ ❝❛s♦✱ ❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ é ✐♥❞❡t❡r♠✐♥❛❞❛✱ ♦✉ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♣❛r❡s ♦r❞❡♥❛❞♦s (x, y) q✉❡ s♦❧✉❝✐♦♥❛♠ ❡ss❡ s✐st❡♠❛✳
❆♦ ❢❛③❡r♠♦s ✉♠❛ ❛♥❛❧♦❣✐❛ ❝♦♠ ♦ ❡st✉❞♦ ❞❛s r❡t❛s✱ ♣❡r❝❡❜❡♠♦s q✉❡ ❡ss❡s ♣❛r❡s ♦r✲ ❞❡♥❛❞♦s q✉❡ s♦❧✉❝✐♦♥❛♠ ♦ s✐st❡♠❛ ❞❡✈❡♠ ❡st❛r ❝♦♥t✐❞♦s ♥❛s ❞✉❛s r❡t❛s✱ ❡ ♣❛r❛ q✉❡ ❞✉❛s r❡t❛s ❝♦♥t❡♥❤❛♠ ♠❛✐s ❞❡ ✉♠ ♣❛r ♦r❞❡♥❛❞♦ ❡♠ ❝♦♠✉♠✱ s❛❜❡♠♦s q✉❡ ❡ss❛s r❡t❛s ❞❡✈❡♠ s❡r ❝♦✐♥❝✐❞❡♥t❡s✱ ♦✉ s❡❥❛✱ ❛s ❞✉❛s ❡q✉❛çõ❡s ❞♦ s✐st❡♠❛✱ ♠❡s♠♦ q✉❡ ❝♦♠ ❛r❣✉♠❡♥t♦s ❞✐❢❡r❡♥t❡s✱ r❡♣r❡s❡♥t❛♠ ❛ ♠❡s♠❛ r❡t❛ ♥♦ ♣❧❛♥♦ ❝♦♠♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ♥❛ ✜❣✉r❛ ✷✳✷
❋✐❣✉r❛ ✷✳✷✿ r❡t❛s ❝♦✐♥❝✐❞❡♥t❡s
P♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ q✉❛❧q✉❡r ♣♦♥t♦ q✉❡ ❡st❡❥❛ ♥❛ r❡t❛✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦P1✱P2✱
P3✱ sã♦ s♦❧✉çõ❡s ❞❡ss❡ s✐st❡♠❛✱ ❡ ❝♦♠♦ ❡①✐st❡♠ ✐♥✜♥✐t♦s ♣♦♥t♦s ❡♠ ✉♠❛ r❡t❛✱ ❡①✐st❡♠
t❛♠❜é♠ ✐♥✜♥✐t❛s s♦❧✉çõ❡s ♣❛r❛ ❡st❡ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s✳
✸♦ ❈❆❙❖ ✲ (Det= 0, Det
x 6= 0 Dety 6= 0)
❈♦♠♦ ❥❛ ✈✐♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ♥❡st❡ ❝❛s♦✱ ❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ é ✐♠♣♦ssí✈❡❧✱ ♦✉ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ♣❛r ♦r❞❡♥❛❞♦s(x, y) q✉❡ s♦❧✉❝✐♦♥❡ ❡ss❡ s✐st❡♠❛✳
❆♦ ❢❛③❡r♠♦s ✉♠❛ ❛♥❛❧♦❣✐❛ ❝♦♠ ♦ ❡st✉❞♦ ❞❛s r❡t❛s✱ ♣❡r❝❡❜❡♠♦s q✉❡ ♥❡st❡ ❝❛s♦✱ ♣❛r❛ ♥ã♦ ❡①✐st✐r ♥❡♥❤✉♠ ♣❛r ♦r❞❡♥❛❞♦ q✉❡ ❡st❡❥❛ ❛♦ ♠❡s♠♦ t❡♠♣♦ ♥❛s ❞✉❛s r❡t❛s✱ é ♥❡❝❡ssár✐♦ q✉❡ ❡ss❛s r❡t❛s ♥ã♦ s❡ ✐♥t❡rs❡❝t❡♠ ❡♠ ♥❡♥❤✉♠ ♣♦♥t♦✱ ♦✉ s❡❥❛✱ ❡ss❛s r❡t❛s ❞❡✈❡♠ s❡r ♣❛r❛❧❡❧❛s ❝♦♠♦ ♦❜s❡r✈❛r❡♠♦s ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✳
❋✐❣✉r❛ ✷✳✸✿ r❡t❛s ♣❛r❛❧❡❧❛s
