APLICAÇÕES DE SISTEMAS LINEARES E DETERMINANTES NA ENGENHARIA CIVIL

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ▼❆❚❖ ●❘❖❙❙❖ ❉❖ ❙❯▲ ■◆❙❚■❚❯❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲

❊▲❚❖◆ ❉❆ ❙■▲❱❆ P❆■❱❆ ❱❆▲■❊◆❚❊

❆P▲■❈❆➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ▲■◆❊❆❘❊❙ ❊

❉❊❚❊❘▼■◆❆◆❚❊ ◆❆ ❊◆●❊◆❍❆❘■❆ ❈■❱■▲

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ▼❆❚❖ ●❘❖❙❙❖ ❉❖ ❙❯▲ ■◆❙❚■❚❯❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲

❊▲❚❖◆ ❉❆ ❙■▲❱❆ P❆■❱❆ ❱❆▲■❊◆❚❊

❆P▲■❈❆➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ▲■◆❊❆❘❊❙ ❊

❉❊❚❊❘▼■◆❆◆❚❊ ◆❆ ❊◆●❊◆❍❆❘■❆ ❈■❱■▲

❖r✐❡♥t❛❞♦r❛✿ Pr♦❢✳➟ ❉r✳➟ ❘ú❜✐❛ ▼❛r❛ ❞❡ ❖❧✐✈❡✐r❛ ❙❛♥t♦s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡s✲ tr❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐✲ t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ■◆▼❆✴❯❋▼❙✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡✳

❈❆▼P❖ ●❘❆◆❉❊✲▼❙ ❙❊❚❊▼❇❘❖ ❉❊ ✷✵✶✺

❆P▲■❈❆➬Õ❊❙ ❉❊ ❙■❙❚❊▼❆❙ ▲■◆❊❆❘❊❙ ❊

❉❊❚❊❘▼■◆❆◆❚❊ ◆❆ ❊◆●❊◆❍❆❘■❆ ❈■❱■▲

❊▲❚❖◆ ❉❆ ❙■▲❱❆ P❆■❱❆ ❱❆▲■❊◆❚❊

❉✐ss❡rt❛çã♦ s✉❜♠❡t✐❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼❛t♦ ●r♦ss♦ ❞♦ ❙✉❧✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡✳

❆♣r♦✈❛❞♦ ♣❡❧❛ ❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛✿

Pr♦❢✳➟ ❉r✳➟ ❘ú❜✐❛ ▼❛r❛ ❞❡ ❖❧✐✈❡✐r❛ ❙❛♥t♦s ✲ ❯❋▼❙ ✭♦r✐❡♥t❛❞♦r❛✮ Pr♦❢✳ ❉r✳ ❆♥❞rés ❇❛t✐st❛ ❈❤❡✉♥❣ ✲ ❯❋▼❙

Pr♦❢✳➟ ❉r✳➟ ❙✐❧✈✐❛ ▲♦♣❡s ❞❡ ❙❡♥❛ ❚❛❣❧✐❛❧❡♥❤❛ ✲ ❯❋❙❈

❈❆▼P❖ ●❘❆◆❉❊✲▼❙ ❙❊❚❊▼❇❘❖ ❉❊ ✷✵✶✺

❉❡❞✐❝♦ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣❡❧♦ ❛♣♦✐♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧✱ ✐♥❝❡♥t✐✈♦ ❡ ❝♦♠♣r❡✲ ❡♥sã♦ s❡♠ ✐❣✉❛❧ ♣❡❧❛ ♠✐♥❤❛ ❛✉sê♥❝✐❛ ❡♥q✉❛♥t♦ ❞❡s❡♥✈♦❧✈✐❛ ❡st❛ ❞✐ss❡rt❛✲ çã♦✳

❊♣í❣r❛❢❡

✧◆ã♦ ❤á ♥❛❞❛ ♠❛✐s ♣rát✐❝♦ ❞♦ q✉❡ ✉♠❛ ❜♦❛ t❡♦r✐❛✳✧

❲✐❧❧✐❛♠ ❏❛♠❡s

❆●❘❆❉❊❈■▼❊◆❚❖❙

❆❣r❛❞❡ç♦ ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r ❛ ❉❡✉s✱ ♣♦r t❡r ♠❡ ❞❛❞♦ ♣❛❝✐ê♥❝✐❛ ❡ s❛❜❡❞♦r✐❛ ♣❛r❛ ❜✉s❝❛r ♠❡✐♦s ❞❡ ❞❡s❡♥✈♦❧✈❡r ❡st❡ tr❛❜❛❧❤♦✳

❊♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ❣❡♥❡r♦s❛ ❡s♣♦s❛ ❱❛♥❡ss❛ ♣♦r ❡st❛r s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦ ♠❡ ❛♣♦✐❛♥❞♦ ❡♥q✉❛♥t♦ ♣❡sq✉✐s❛✈❛ ❡ ❞❡s❡♥✈♦❧✈✐❛ ❡st❡ tr❛❜❛❧❤♦✳

❆s ♠✐♥❤❛s ❧✐♥❞❛s ✜❧❤❛s✱ ■s❛❜❡❧❛ ❡ ❇❡❛tr✐③✱ ♣♦r ❡♥t❡♥❞❡r❡♠ ❛ ❢❛❧t❛ ❞❡ t❡♠♣♦ ♣❛r❛ ❜r✐♥❝❛r✳

❆♦ ♠❡✉ ♣❛✐ ◆♦r❜❡rt♦✱ ♠✐♥❤❛ ♠ã❡ ❖❞á❧✐❛✱ ♠❡✉ ✐r♠ã♦ ❊♠❡rs♦♥✱ ♣❡❧❛ ❛t❡♥çã♦ ❞❛❞❛ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛ ❡♥q✉❛♥t♦ tr❛❜❛❧❤❛✈❛✳

❆♦ ♠❡✉ s♦❣r♦ ◆❛r❝✐s♦✱ ♠✐♥❤❛ s♦❣r❛ ▼❛r✐❛ ■♥ês✱ ♣♦r ❝✉✐❞❛r❡♠ ❞❛s ♠✐♥❤❛s ✜❧❤❛s q✉❛♥❞♦ ♣r❡❝✐s❛✈❛✳

❆ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ Pr♦❢➟ ❉r➟ ❘ú❜✐❛ ♣❡❧❛ ❛t❡♥çã♦ ❡ ♣❛❝✐ê♥❝✐❛ q✉❡ ♠❡ ❝♦♥❝❡❞❡✉ ♥❛s ✈ár✐❛s r❡✉♥✐õ❡s ❡ tr♦❝❛s ❞❡ ♠❡♥s❛❣❡♥s q✉❡ t✐✈❡♠♦s ♣❛r❛ tr❛t❛r ❞❡st❛ ❞✐ss❡rt❛çã♦✳

❆♦ Pr♦❢✳ ❉r✳ ❆♥❞rés ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ❛♣♦✐♦ ❞❛❞♦ ♣❛r❛ q✉❡ ❡ss❡ tr❛❜❛❧❤♦ ❢♦ss❡ ❜❡♠ ❞❡s❡♥✈♦❧✈✐❞♦✱ ✐♥❞✐❝❛♥❞♦ ❜✐❜❧✐♦❣r❛✜❛ ❡ ❛✉①✐❧✐❛♥❞♦ ♥❛ ❡s❝r✐t❛✳

❊♠ s✉♠❛✱ ❛❣r❛❞❡ç♦ ❛ t♦❞♦s q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐✲ ♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❘❡s✉♠♦

❖ ❝♦♥t❡ú❞♦ ❞❡st❛ ❞✐ss❡rt❛çã♦ ❛❜♦r❞❛ ❛♣❧✐❝❛çõ❡s ❞♦s ❝♦♥❝❡✐t♦s ▼❛tr✐③❡s✱ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❡ ❉❡t❡r♠✐♥❛♥t❡ ❡♠ ♣r♦❜❧❡♠❛s r❡❛✐s ♥❛ ❊♥❣❡♥❤❛r✐❛ ❈✐✈✐❧✱ t❛✐s ❝♦♠♦✿ r❡s♦❧✉çã♦ ❞❡ tr❡❧✐ç❛s ❡ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦s ♠♦❞♦s ❞❡ ✈✐❜r❛çã♦ ❞❡ ✉♠❛ ❡str✉t✉r❛✱ s❡♥❞♦ ❡st❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡❧❡♠❡♥t♦s ❝♦♥❡❝t❛❞❛s ♣❛r❛ s✉♣♦rt❛r ✉♠ ❝❛rr❡❣❛♠❡♥t♦✳ ❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ é ❢❡✐t♦ ✐♥✐❝✐❛❧♠❡♥t❡ ❝♦♠ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❤✐stór✐❝❛ ❡ ♣♦st❡r✐♦r♠❡♥t❡ é ❛♣r❡s❡♥t❛❞❛ ❛ ❢✉♥❞❛✲ ♠❡♥t❛çã♦ t❡ór✐❝❛✱ ❡①♣♦♥❞♦ ❛s ❞❡✜♥✐çõ❡s✱ t❡♦r❡♠❛s ❡ ♣r♦♣r✐❡❞❛❞❡s✳ P❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s✱ sã♦ ❡①✐❜✐❞♦s ▼ét♦❞♦s ❉✐r❡t♦s ❡ ■t❡r❛t✐✈♦s✱ s❡♥❞♦ ❡st❡ ú❧t✐♠♦ ✉s❛❞♦ ❡♠ ♣r♦❜❧❡♠❛s r❡❛✐s ♣♦r s❡ tr❛t❛r ❞❡ s✐t✉❛çõ❡s q✉❡ ❡♥✈♦❧✈❡♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ✐♥❝ó❣♥✐t❛s ❡ ❡q✉❛çõ❡s✳ ❈♦♥t✉❞♦✱ ♣❛r❛ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦ ❞❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❝♦♥✲ ❝❡✐t♦s ❢ís✐❝♦s ❝♦♠♦ ❡q✉✐❧í❜r✐♦ ❡stát✐❝♦ ❞❡ ✉♠ ❝♦r♣♦ ❡ r❡s✉❧t❛❞♦s ❥á ❝♦♥❤❡❝✐❞♦s ❞❡ ✈✐❜r❛çõ❡s✳

P❛❧❛✈r❛s✲❈❤❛✈❡✿ ❙✐st❡♠❛s ▲✐♥❡❛r❡s✳ ❉❡t❡r♠✐♥❛♥t❡✳ ❚r❡❧✐ç❛s✳ ❱✐❜r❛çõ❡s✳

❆❜str❛❝t

❚❤❡ ❝♦♥t❡♥t ♦❢ t❤✐s t❤❡s✐s ❝♦♠♣r❡❤❡♥❞s t❤❡ ❝♦♥❝❡♣ts ♦❢ ▼❛tr✐❝❡s✱ ▲✐♥❡❛r ❙②st❡♠s ❛♥❞ ❉❡t❡r♠✐♥❛♥ts ❛♣♣❧✐❡❞ t♦ ♣r♦❜❧❡♠ s♦❧✈✐♥❣ ♦❢ r❡❛❧ s✐t✉❛t✐♦♥s ✐♥ ❈✐✈✐❧ ❊♥❣✐♥❡❡r✐♥❣✱ s✉❝❤ ❛s✿ tr✉ss❡s r❡s♦❧✉t✐♦♥s ❛♥❞ t❤❡ ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ ✈✐❜r❛t✐♦♥ ♠♦❞❡s ♦❢ ❛ str✉❝t✉r❡✱ ❝♦♥s✐❞❡✲ r✐♥❣ t❤✐s ❛s ❛ ❣r♦✉♣ ♦❢ ❛ss❡♠❜❧❡❞ ❡❧❡♠❡♥ts ✐♥ ♦r❞❡r t♦ s✉♣♣♦rt ❛ ❧♦❛❞✳ ❚❤❡ ❞❡✈❡❧♦♣♠❡♥t ✐s ✐♥✐t✐❛❧❧② ❝❛rr✐❡❞ ♦✉t t❤r♦✉❣❤ ❛ ❤✐st♦r✐❝❛❧ ❛♣♣r♦❛❝❤✳ ❆❢t❡r t❤❛t✱ t❤❡ t❤❡♦r❡t✐❝❛❧ ❜❛s✐s ✐s ✐♥tr♦❞✉❝❡❞✱ s❤♦✇✐♥❣ ❞❡✜♥✐t✐♦♥s✱ t❤❡♦r❡♠s ❛♥❞ ♣r♦♣❡rt✐❡s✳ ❋♦r ▲✐♥❡❛r ❙②st❡♠s r❡s♦❧✉t✐✲ ♦♥s✱ ❉✐r❡❝t ❛♥❞ ■♥t❡r❛❝t✐✈❡ ▼❡t❤♦❞s ❛r❡ ♣r❡s❡♥t❡❞✱ ❜❡✐♥❣ t❤❡ ❧❛tt❡r ✉s❡❞ ✐♥ r❡❛❧ ♣r♦❜❧❡♠ s♦❧✈✐♥❣ s✐♥❝❡ t❤❡② ❛r❡ s✐t✉❛t✐♦♥s t❤❛t ❝♦♠♣r❡❤❡♥❞ s❡✈❡r❛❧ ✉♥❦♥♦✇♥ ❢❛❝t♦rs ❛♥❞ ❡q✉❛t✐♦♥s✳ ◆❡✈❡rt❤❡❧❡ss✱ ❢♦r ❛ ❜❡tt❡r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❛♣♣❧✐❝❛❜✐❧✐t②✱ ♣❤②s✐❝❛❧ ❝♦♥❝❡♣ts s✉❝❤ ❛s t❤❡ st❛t✐❝ ❜❛❧❛♥❝❡ ♦❢ ❛ ❜♦❞② ❛♥❞ ✈✐❜r❛t✐♦♥ ❦♥♦✇♥ r❡s✉❧ts ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞✳

❑❡②✇♦r❞s✿ ▲✐♥❡❛r ❙②st❡♠s✳ ❉❡t❡r♠✐♥❛♥ts✳ ❚r✉ss❡s✳ ❱✐❜r❛t✐♦♥s✳

◆♦♠❡♥❝❧❛t✉r❛

A= (aij) ▼❛tr✐③ ❢♦r♠❛❞❛ ♣♦r ❡❧❡♠❡♥t♦s aij ❞❡ ♦r❞❡♠ m×n✱ ❝♦♠

i= 1,2, .., m ❡ j = 1,2, ..., n

Mm×n(R) ❈♦♥❥✉♥t♦ ❞❛s ♠❛tr✐③❡s r❡❛✐s ❞❡ ♦r❞❡♠ m×n

Mn(R) ❈♦♥❥✉♥t♦s ❞❛s ♠❛tr✐③❡s r❡❛✐s q✉❛❞r❛❞❛s ❞❡ ♦r❞❡♠ n

Am×n ▼❛tr✐③ ❞❡ ♦r❞❡♠ m×n✱ ♦✉ s❡❥❛✱ ❝♦♠ ♠ ❧✐♥❤❛s ❡ ♥ ❝♦❧✉♥❛s

A−1 _{▼❛tr✐③ ■♥✈❡rs❛ ❞❡ A✱AA}−1 _=A−1_{A =I}

n

In ▼❛tr✐③ ■❞❡♥t✐❞❛❞❡ ❞❡ ♦r❞❡♠ n✱ aii= 1 ❡ aij = 0 s❡ i6=j

¯

A ▼❛tr✐③ ❞♦s ❝♦❢❛t♦r❡s adj A ▼❛tr✐③ ❆❞❥✉♥t❛

F ▼❛tr✐③ ❞❛s ❋♦rç❛s✱ ❈❛rr❡❣❛♠❡♥t♦s K ▼❛tr✐③ ❞❡ ❘✐❣✐❞❡③

X ▼❛tr✐③ ❞❡ ❉❡s❧♦❝❛♠❡♥t♦s M ▼❛tr✐③ ❞❡ ▼❛ss❛

C ▼❛tr✐③ ❞❡ ❆♠♦rt❡❝✐♠❡♥t♦ AT _{❚r❛♥s♣♦st❛ ❞❡ A}

Aij ❙✉❜♠❛tr✐③ ❞❡ A✱ ♥❛ q✉❛❧ ❛ ✐✲és✐♠❛ ❧✐♥❤❛ ❡ ❛ ❥✲és✐♠❛ ❝♦❧✉♥❛ ❢♦r❛♠ r❡t✐r❛❞❛s

△ij ❈♦❢❛t♦r ♦✉ ❝♦♠♣❧❡♠❡♥t♦ ❛❧❣é❜r✐❝♦ ❞♦ ❡❧❡♠❡♥t♦ aij

k ◆ú♠❡r♦ r❡❛❧

n

P

k=1

❙♦♠❛tór✐♦ ❞❡✜♥✐❞♦ ❝♦♠ k ✈❛r✐❛♥❞♦ ❞❡ 1 ❛té n Li ▲✐♥❤❛ i ❞❡ ✉♠❛ ♠❛tr✐③ q✉❛❧q✉❡r

e(In) ▼❛tr✐③ ♦❜t✐❞❛ ❞❛ ■❞❡♥t✐❞❛❞❡ ♣♦r ♠❡✐♦ ❞❡ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s

(α1, α2, ..., αn) ▲✐st❛ ♦r❞❡♥❛❞❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s ♦✉ s✐♠♣❧❡s♠❡♥t❡ vetor

Rn _{❈♦♥❥✉♥t♦ ❞❛s n−uplas ❞❡ ♥ú♠❡r♦s r❡❛✐s}

kxk ◆♦r♠❛ ❞❡ ✉♠ ✈❡t♦r

kAk ◆♦r♠❛ ❞❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ |xi| ▼ó❞✉❧♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ xi

x(k) _{❙❡q✉ê♥❝✐❛ ❞❡ ✈❡t♦r❡s}

det A ♦✉|A| ❉❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A

ΣFx ❙♦♠❛tór✐♦ ❞❡ t♦❞❛s ❛s ❢♦rç❛s ❝♦♠♣♦♥❡♥t❡ ❡♠ x

ΣFy ❙♦♠❛tór✐♦ ❞❡ t♦❞❛s ❛s ❢♦rç❛s ❝♦♠♣♦♥❡♥t❡ ❡♠ y

ΣM ❙♦♠❛tór✐♦ ❞❡ ♠♦♠❡♥t♦ ❞❡✈✐❞♦s ❛s ❢♦rç❛s ❡♠ r❡❧❛çã♦ ❛ ✉♠ ♣♦♥t♦

q✉❛❧q✉❡r

+

→ ❙❡♥t✐❞♦ ❛❞♦t❛❞♦ ❝♦♠♦ ♣♦s✐t✐✈♦ ♣❛r❛ ❡✐①♦ x +↑ ❙❡♥t✐❞♦ ❛❞♦t❛❞♦ ❝♦♠♦ ♣♦s✐t✐✈♦ ♣❛r❛ ❡✐①♦ y

+

ց ❙❡♥t✐❞♦ ❛❞♦t❛❞♦ ❝♦♠♦ ♣♦s✐t✐✈♦ ♣❛r❛ ❡✐①♦ x +ր ❙❡♥t✐❞♦ ❛❞♦t❛❞♦ ❝♦♠♦ ♣♦s✐t✐✈♦ ♣❛r❛ ❡✐①♦ y +տ ❙❡♥t✐❞♦ ❛❞♦t❛❞♦ ❝♦♠♦ ♣♦s✐t✐✈♦ ♣❛r❛ ❡✐①♦ y

θA ❘♦t❛çã♦ ♥♦ ♣♦♥t♦ A

∆B ❉❡s❧♦❝❛♠❡♥t♦ ♥♦ ♣♦♥t♦ B

ω ❋r❡q✉ê♥❝✐❛ ❝✐r❝✉❧❛r f ❋r❡q✉ê♥❝✐❛ ❞❡ ♦s❝✐❧❛çã♦

φn ❱❡t♦r q✉❡ r❡♣r❡s❡♥t❛ ♦s ♠♦❞♦s ❞❡ ✈✐❜r❛çã♦ ♥❛t✉r❛❧ ❞❛ ❡str✉t✉r❛

v(t) ❉❡s❧♦❝❛♠❡♥t♦ ❞♦ s✐st❡♠❛ ❡♠ ❞✐r❡çã♦ à ✉♠ ❡✐①♦ ❞❡ r❡❢❡rê♥❝✐❛ ˙

v(t) ❱❡❧♦❝✐❞❛❞❡ ❞♦ s✐st❡♠❛ ¨

v(t) ❆❝❡❧❡r❛çã♦ ❞♦ s✐st❡♠❛

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✹✳✶ ▼♦❞❡❧♦ ❞❡ ♣♦♥t❡ tr❡❧✐ç❛❞❛ ❬✶✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✷ P❧❛❝❛ ❞❡ r❡❢♦rç♦ ✭❛✮ ❡ ▲✐❣❛çã♦ ❝♦♠ ♣✐♥♦ ✭❜✮ ❬✶✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✸ ❚r❡❧✐ç❛s P❧❛♥❛s ✭❛✮ ❡ ❊s♣❛❝✐❛✐s ✭❜✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✹ ❊str✉t✉r❛ ✐♥stá✈❡❧ ✭❛✮ ❡ ❊str✉t✉r❛ ❡stá✈❡❧ ✭❜✮ ❬✶✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✺ ❚r❡❧✐ç❛ ❙✐♠♣❧❡s ❬✶✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✻ ❚r❡❧✐ç❛s ❈♦♠♣♦st❛s ❬✶✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✼ ❚r❡❧✐ç❛ ❈♦♠♣❧❡①❛ ❬✶✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✽ ❚r❡❧✐ç❛s ♣❧❛♥❛s ✉s❛❞❛s ❡♠ t❡❧❤❛❞♦s ❬✶✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✾ ❚r❡❧✐ç❛s ♣❧❛♥❛s ✉s❛❞❛s ❡♠ ♣♦♥t❡s ❬✶✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✶✵ ❊str✉t✉r❛ ❞❡ t❡❧❤❛❞♦ ✭❛✮ ❡ ♠♦❞❡❧♦ ❞❡ tr❡❧✐ç❛ ✲ t❡s♦✉r❛ ✭❜✮ ❬✶✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✶✶ ❊str✉t✉r❛ ❞❡ ♣♦♥t❡ ✭❛✮ ❡ ♠♦❞❡❧♦ ❞❡ tr❡❧✐ç❛ ❞❡ ♣♦♥t❡ ✭❜✮ ❬✶✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✶✷ ❉✐❛❣r❛♠❛s ❞❡ ❝♦r♣♦ ❧✐✈r❡ ❞❡ ❜❛rr❛s ❝❛rr❡❣❛❞❛s ❛①✐❛❧♠❡♥t❡ ❡ ♥ós ❛❞❥❛❝❡♥t❡s✿

✭❛✮ ❜❛rr❛ ❆❇ ❡♠ tr❛çã♦❀ ✭❜✮ ❜❛rr❛ ❆❇ ❡♠ ❝♦♠♣r❡ssã♦ ❬✷✵❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✶✸ ❚r❡❧✐ç❛s ✐♥stá✈❡✐s ❬✶✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✶✹ ❚r❡❧✐ç❛ ❡stá✈❡❧ ❬✶✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✶✺ ❚r❡❧✐ç❛ ✐♥t❡r♥❛♠❡♥t❡ ✐♥stá✈❡❧ ❬✶✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✶✻ ❚r❡❧✐ç❛ ✶ ❬✶✽❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✹✳✶✼ ❋♦rç❛s ❛t✉❛♥t❡s ♥♦ ♥ó ❆ ✭❛✮✱ ♥ó ❇ ✭❜✮ ❡ ♥ó ❈ ✭❝✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✶✽ ❚r❡❧✐ç❛ ❋✐♥❦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✶✾ ❘❡❛çõ❡s ❞♦s ❛♣♦✐♦s ❡ ❛rt✐❝✉❧❛çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✹✳✷✵ ❋♦rç❛s ❛♣❧✐❝❛❞❛s ✲ ◆ó ❆✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✹✳✷✶ ❋♦rç❛s ❛♣❧✐❝❛❞❛s ✲ ◆ó ❇✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✹✳✷✷ ❋♦rç❛s ❛♣❧✐❝❛❞❛s ✲ ◆ó ❈✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✹✳✷✸ ❋♦rç❛s ❛♣❧✐❝❛❞❛s ✲ ◆ó ❉✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✹✳✷✹ ❋♦rç❛s ❛♣❧✐❝❛❞❛s ✲ ◆ó ❊ ❡ ❡sq✉❡♠❛ ❣❡♦♠étr✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✹✳✷✺ ❋♦rç❛s ❛♣❧✐❝❛❞❛s ✲ ◆ó ❋ ❡ ❡sq✉❡♠❛ ❣❡♦♠étr✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✷✻ ❋♦rç❛s ❛♣❧✐❝❛❞❛s ✲ ◆ó ● ❡ ❡sq✉❡♠❛ ❣❡♦♠étr✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✷✼ ❘❡❛çõ❡s ❞❡ ❛♣♦✐♦ ❡ ❡s❢♦rç♦s ♥❛s ❜❛rr❛s ✭❝♦rr✐❣✐❞♦s✮✱ ❡♠ ❦◆✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✷✽ ❊str✉t✉r❛s ❝♦♠ ✉♠ ✭❛✮✱ q✉❛tr♦ ✭❜✮ ❡ três ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ✭❝✮ ❬✷✶❪ ✳ ✳ ✳ ✳ ✻✺ ✹✳✷✾ ❱✐❜r❛çõ❡s ❧✐✈r❡ ✭❛✮ ❡ ❢♦rç❛❞❛ ✭❜✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✹✳✸✵ P♦♥t❡ ❚❛❝♦♠❛ ◆❛rr♦✇s ✲ ❊❯❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✹✳✸✶ Pórt✐❝♦ ❞❡ três ♥í✈❡✐s ✭❛✮ ❡ ❈♦❡✜❝✐❡♥t❡ ❞❡ r✐❣✐❞❡③ ❡♠ ❝❛❞❛ ♥ó ❞♦ ♣órt✐❝♦ ✭❜✮ ✻✾ ✹✳✸✷ ▼♦❞♦s ❞❡ ✈✐❜r❛çã♦ ❞❛ ❡str✉t✉r❛ ❞❡❢♦r♠❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶

✶✳✶ ❆❜♦r❞❛❣❡♠ ❍✐stór✐❝❛ s♦❜r❡ ▼❛tr✐③❡s✱ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❡ ❉❡t❡r♠✐♥❛♥t❡ ✳ ✷ ✶✳✷ ❖ ❡♥s✐♥♦ ❞❡ ▼❛tr✐③❡s ❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ♥❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ ❖❜❥❡t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✷ ▼❛tr✐③❡s ❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ✼

✷✳✶ ❉❡✜♥✐çõ❡s ❇ás✐❝❛s ❡ ❚✐♣♦s ❞❡ ▼❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✷ ❖♣❡r❛çõ❡s ❝♦♠ ▼❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✸ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸✳✶ ❋♦r♠❛ ▼❛tr✐❝✐❛❧ ❞❡ ✉♠ ❙✐st❡♠❛ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✸✳✷ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❊q✉✐✈❛❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸✳✸ ❈❧❛ss✐✜❝❛çã♦ ❞❡ ✉♠ ❙✐st❡♠❛ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✸✳✹ ❙✐st❡♠❛s ❊s❝❛❧♦♥❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸✳✺ ❉✐s❝✉ssã♦ ❡ ❘❡s♦❧✉çã♦ ❞❡ ✉♠ ❙✐st❡♠❛ ▲✐♥❡❛r ♣♦r ❊s❝❛❧♦♥❛♠❡♥t♦ ✳ ✳ ✷✶ ✷✳✹ ▼ét♦❞♦s ■t❡r❛t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✹✳✶ ▼ét♦❞♦ ❞❡ ❏❛❝♦❜✐✲❘✐❝❤❛r❞s♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✹✳✷ ▼é❞♦❞♦ ❞❡ ●❛✉ss✲❙❡✐❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✷✳✺ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✸ ❉❡t❡r♠✐♥❛♥t❡ ❡ ▼❛tr✐③ ■♥✈❡rs❛ ✸✷

✸✳✶ ❉❡t❡r♠✐♥❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✷ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ▲❛♣❧❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✸ ❖❜t❡♥çã♦ ❞❛ ▼❛tr✐③ ■♥✈❡rs❛ ♣♦r ❉❡t❡r♠✐♥❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✹ Pr♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ❛ ■♥✈❡rsã♦ ❞❡ ▼❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✺ ❆♣❧✐❝❛çã♦ ❞❡ ❉❡t❡r♠✐♥❛♥t❡ ❡♠ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ✲ ❘❡❣r❛ ❞❡ ❈r❛♠❡r ✳ ✳ ✳ ✸✽ ✸✳✻ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✹ ❆♣❧✐❝❛çõ❡s ❞❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❡ ❉❡t❡r♠✐♥❛♥t❡ ✲ ❚r❡❧✐ç❛s ❡ ❱✐❜r❛çõ❡s ✹✶ ✹✳✶ ❚r❡❧✐ç❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✶✳✶ ❈❧❛ss✐✜❝❛çã♦ q✉❛♥t♦ à ❢♦r♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✶✳✷ ❈❧❛ss✐✜❝❛çã♦ q✉❛♥t♦ à ❡st❛t✐❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✶✳✸ ❘❡s♦❧✈❡♥❞♦ ✉♠❛ ❚r❡❧✐ç❛ ■s♦stát✐❝❛ P❧❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✷ ❆♣❧✐❝❛çã♦ ❞❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ✲ ❚r❡❧✐ç❛ ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✸ ❆♣❧✐❝❛çã♦ ❞❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ✲ ❚r❡❧✐ç❛ ❋✐♥❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✹ ❈❛s♦ ●❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✹✳✺ ❆♣❧✐❝❛çõ❡s ❞❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❡ ❉❡t❡r♠✐♥❛♥t❡ ❡♠ ❱✐❜r❛çõ❡s ▲✐✈r❡s ✳ ✳ ✳ ✻✶ ✹✳✺✳✶ ❆✉t♦✈❛❧♦r❡s ❡ ❆✉t♦✈❡t♦r❡s ♥❛ ❊♥❣❡♥❤❛r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✹✳✺✳✷ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ❱✐❜r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✹✳✺✳✸ ❆♣❧✐❝❛çã♦ ❞❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❡ ❉❡t❡r♠✐♥❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✹✳✻ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷

✺ ❈♦♥❝❧✉sã♦ ✼✸

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

➱ s❛❜✐❞♦ q✉❡ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ é ♥❡❝❡ssár✐♦ ❡♠ ❞✐✈❡rs❛s s✐t✉❛çõ❡s s❡❥❛ ❝♦♠♦ ❛♣♦✐♦ ❛ ♦✉tr❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ❝♦♠♦ ✐♥str✉♠❡♥t♦ ♣❛r❛ ❧✐❞❛r ❝♦♠ s✐t✉❛çõ❡s ❞❛ ✈✐❞❛ ❝♦t✐❞✐❛♥❛ ♦✉✱ ❛✐♥❞❛✱ ❝♦♠♦ ❢♦r♠❛ ❞❡ ❞❡s❡♥✈♦❧✈❡r ❤❛❜✐❧✐❞❛❞❡s ❡ ♣❡♥s❛♠❡♥t♦s ❬✶❪✳

◆❛ ❡t❛♣❛ ✜♥❛❧ ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✱ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❛ ▼❛t❡♠át✐❝❛ ❞❡✈❡ s❡r tr❛✲ t❛❞❛ ❝♦♠♦ ✉♠❛ ♣❛r❝❡❧❛ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ss❡♥❝✐❛❧ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞❡ t♦❞♦s ♦s ❥♦✈❡♥s✱ s❡ ❢❛③❡♥❞♦ r❡s♣♦♥sá✈❡❧ ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡ ✈✐sã♦ ❞❡ ♠✉♥❞♦✱ ♣❛r❛ ❧❡r ❡ ✐♥t❡r♣r❡t❛r ❛ r❡❛❧✐❞❛❞❡ ❡ ❞❡s❡♥✈♦❧✈❡r ❝❛♣❛❝✐❞❛❞❡s q✉❡ ❞❡❧❡s s❡rã♦ ❡①✐❣✐❞❛s ❛♦ ❧♦♥❣♦ ❞❛ ✈✐❞❛ s♦❝✐❛❧ ❡ ♣r♦✜ss✐♦♥❛❧✳ P♦rt❛♥t♦✱ ❛ ▼❛t❡♠át✐❝❛ t❡♠ ♦ ♣♦❞❡r ❞❡ ✐r ❛❧é♠ ❞❡ s❡✉ ❝❛rát❡r ✐♥str✉♠❡♥t❛❧✱ ❝❛r❛❝t❡r✐③❛♥❞♦✲s❡ ❝♦♠♦ ❝✐ê♥❝✐❛ ❝♦♠ ❝❛r❛❝t❡ríst✐❝❛s ♣ró♣r✐❛s ❞❡ ✐♥✈❡st✐❣❛çã♦ ❛tr❛✈és ❞❡ ❧✐♥✲ ❣✉❛❣❡♠ ♣ró♣r✐❛ ❡ ❝♦♠ ♣❛♣❡❧ ❡ss❡♥❝✐❛❧ ❥✉♥t♦ às ❞❡♠❛✐s ❈✐ê♥❝✐❛s ❞❛ ◆❛t✉r❡③❛✳ ❙♦❜ ♦ ♦❧❤❛r ❞❡ ❝✐ê♥❝✐❛✱ s✉❛ ❞✐♠❡♥sã♦ ❤✐stór✐❝❛✱ s✉❛ í♥t✐♠❛ r❡❧❛çã♦ ❝♦♠ ❛ s♦❝✐❡❞❛❞❡✱ ❛ ❝✉❧t✉r❛ ❞❡ ❞✐❢❡✲ r❡♥t❡s é♣♦❝❛s ❛♠♣❧✐❛♠ ❡ ❛♣r♦❢✉♥❞❛♠ ♦ ❡s♣❛ç♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s ♥ã♦ só ♥❡st❛ ❞✐s❝✐♣❧✐♥❛✱ ♠❛s ♠♦♥str❛♥❞♦ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♥❛s ✐♥t❡r✲r❡❧❛çõ❡s ❝♦♠ ♦✉tr❛s ár❡❛s ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❬✶❪✳

❈♦♥s✐❞❡r❛♥❞♦ ❛ ❢♦r♠❛ ❞❡ ❛❜♦r❞❛❣❡♠ ❛♣r❡s❡♥t❛❞❛ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞❡ ❛ss✉♥t♦s ❡s♣❡❝í✜❝♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❡♠ ❡s♣❡❝✐❛❧✱ ▼❛tr✐③❡s✱ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❡ ❉❡t❡r♠✐♥❛♥t❡✱ s❡♥t✐✉✲s❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❛♣r❡s❡♥t❛r ❞❡ ❢♦r♠❛ s✐♠♣❧❡s ❛♣❧✐❝❛çõ❡s ❞❡ss❡s ❝♦♥❝❡✐t♦s✳ ❉❡ss❛ ❢♦r♠❛✱ s❡rá ❞❡s❡♥✈♦❧✈✐❞♦ ✉♠❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ ❞♦s ❝♦♥❝❡✐t♦s s✉♣r❛❝✐t❛❞♦s✱ ♣❛r❛ ♣♦st❡r✐♦r ❛♣❧✐❝❛çã♦ ❡♠ ♣r♦❜❧❡♠❛s ❝♦♥❝r❡t♦s ❞❛ ❊♥❣❡♥❤❛r✐❛ ❈✐✈✐❧✳ ❊ss❛s ❛♣❧✐❝❛çõ❡s s❡rã♦ ❢❡✐t❛s ♥❛s r❡s♦❧✉çõ❡s ❞❡ tr❡❧✐ç❛s ❡ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦s ♠♦❞♦s ❞❡ ✈✐❜r❛çã♦ ❞❡ ✉♠❛ ❡str✉t✉r❛ ❛♣♦rt✐❝❛❞❛ ❞❡ três ♣❛✈✐♠❡♥t♦s✳ P❛r❛ t❛♥t♦✱ s❡ ❢❛rá ♥❡❝❡ssár✐♦ ❛s ❞❡✜♥✐çõ❡s ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❢ís✐❝♦s ❡ ♥♦♠❡♥❝❧❛t✉r❛s ✉s❛❞❛s ♥❛s ❊♥❣❡♥❤❛r✐❛s✳

❖ ❧❡✐t♦r✱ s❡❥❛ ❡❧❡ ✉♠ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ♦✉ ✉♠ ❛❧✉♥♦ ❞❡ ❣r❛❞✉❛çã♦ ❡♠ ❈✐ê♥❝✐❛s ❊①❛t❛s✱ ❡♥❝♦♥tr❛rá ❛q✉✐ ❡①❡♠♣❧♦s ❞❡ ❛♣❧✐❝❛çõ❡s ❞❡ ❝♦♥❝❡✐t♦s ✈✐st♦s ♥❛s ❞✐s❝✐♣❧✐♥❛s ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r ❡ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✳

✶✳✶ ❆❜♦r❞❛❣❡♠ ❍✐stór✐❝❛ s♦❜r❡ ▼❛tr✐③❡s✱ ❙✐st❡♠❛s ▲✐✲

♥❡❛r❡s ❡ ❉❡t❡r♠✐♥❛♥t❡

❯♠❛ ❛♥t✐❣❛ ♦❜r❛ ❝❤✐♥❡s❛ ❝♦♥st❛ q✉❡ ❛ ♦r❞❡♠ ❝r♦♥♦❧ó❣✐❝❛ ❞♦s ❛ss✉♥t♦s ❢♦✐ ❙✐st❡♠❛s ▲✐♥❡❛r❡s✱ ❉❡t❡r♠✐♥❛♥t❡ ❡ ▼❛tr✐③❡s✱ s❡♥❞♦ ❡st❛ ú❧t✐♠❛ ❡st✉❞❛❞❛ ❞❡ ❢♦r♠❛ ♣r♦❢✉♥❞❛ ♣♦r ❛♣r❡s❡♥t❛r s✉❛ á❧❣❡❜r❛ ♠✉✐t♦ ♦r❣❛♥✐③❛❞❛✳ ◆♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ■ ❞❛ ❡r❛ ❝r✐stã✱ ♦s ❝❤✐♥❡s❡s ♣✉❜❧✐❝❛r❛♠ ✉♠ ❧✐✈r♦ ✐♥t✐t✉❧❛❞♦ ❑✬✉✐✲❝❤✬❛♥❣ ❙✉❛♥✲❙❤✉ ✭❖s ◆♦✈❡ ❈❛♣ít✉❧♦s s♦❜r❡ ❛ ❆rt❡ ▼❛t❡♠át✐❝❛✮ ❬✷❪✳ ❊ss❛ ♣r♦❞✉çã♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ♣♦r ♣❛rt❡ ❞♦s ❝❤✐♥❡s❡s s❡ ❞❡✉ ❞✉r❛♥t❡ ♦ ❞❡❝❧í♥✐♦ ❞❡ ♣r♦❞✉çã♦ ♣♦r ♣❛rt❡ ❞♦s ❡✉r♦♣❡✉s✱ s❡♥❞♦ q✉❡ ❡st❡s ✐r✐❛♠ r❡❞❡s❝♦❜r✐r r❡s✉❧t❛❞♦s ❥á ♣r♦❞✉③✐❞♦s ♣❡❧♦s ❝❤✐♥❡s❡s ♠✉✐t♦ ♠❛✐s t❛r❞❡✱ ❞✉r❛♥t❡ ♦✉ ❛♣ós ♦ ❘❡♥❛s❝✐♠❡♥t♦✳

◆❛ ♦❜r❛ ❑✬✉✐✲❝❤✬❛♥❣ ❙✉❛♥✲s❤✉✱ ❡♠ s❡✉ ♦✐t❛✈♦ ❝❛♣ít✉❧♦✱ é ❛❜♦r❞❛❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❙✐s✲ t❡♠❛s ❞❡ ❊q✉❛çõ❡s ▲✐♥❡❛r❡s ❡ ♣r♦❝❡❞✐♠❡♥t♦s ♠❛tr✐❝✐❛✐s✳ ❖ ♠ét♦❞♦ ♣❛r❛ r❡s♦❧✈❡r s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❜❛s❡❛✈❛✲s❡ ❝✉r✐♦s❛♠❡♥t❡ ❡♠ ♦♣❡r❛çõ❡s q✉❡ ❡r❛♠ r❡❛❧✐③❛❞❛s ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❡q✉❡♥♦s ❣r❛✈❡t♦s ❞✐s♣♦st♦s ❡♠ ✉♠❛ ❢♦❧❤❛ ❞❡ ♣❛♣❡❧✱ s❡♥❞♦ q✉❡ ❡st❛ té❝♥✐❝❛ s❡ ❛ss❡♠❡❧❤❛✈❛ ❛♦ ♠ét♦❞♦ ❞❡ ❡❧✐♠✐♥❛çã♦ ❞❡ ●❛✉ss ✭❛♣r❡s❡♥t❛❞♦✱ s♦♠❡♥t❡✱ ♥♦ sé❝✉❧♦ ❳■❳✮✳ ❊ss❛s ♦♣❡r❛çõ❡s ❡r❛♠ ❡❢❡t✉❛❞❛s ❝♦♠ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ❡q✉❛çõ❡s q✉❡ ❝♦♠♣✉♥❤❛♠ ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s✱ ❞❛í ❛ ♦r✐❣❡♠ ❞❛ té❝♥✐❝❛ ❛❞♦t❛❞❛ ❝♦♠♦ ♣r❡❝✉rs♦r❛ ❞❡ ♠ét♦❞♦s ❞❡ s♦❧✉çã♦ ♣r♦♣♦st♦s ❡♠ é♣♦❝❛s ♣♦st❡r✐♦r❡s ❬✸❪✳

❊♠ ❬✹❪✱ ❛♣r❡s❡♥t❛ ✉♠ ♣r♦❜❧❡♠❛ ❞❛ é♣♦❝❛ q✉❡ ❢♦✐ r❡s♦❧✈✐❞♦ ✉t✐❧✐③❛❞♦ ❛ té❝♥✐❝❛ ❝❤✐♥❡s❛✱ ❛ s❛❜❡r✿

❼ ✏❚rês ❢❡✐①❡s ❞❡ ✉♠❛ ❝♦❧❤❡✐t❛ ❞❡ ❜♦❛ q✉❛❧✐❞❛❞❡✱ ❞♦✐s ❢❡✐①❡s ❞❡ ✉♠❛ ❞❡ q✉❛❧✐❞❛❞❡ r❡❣✉❧❛r ❡ ✉♠ ❢❡✐①❡ ❞❡ ✉♠❛ ❞❡ ♠á q✉❛❧✐❞❛❞❡ sã♦ ✈❡♥❞✐❞♦s ♣♦r ✸✾ ❞♦✉✳ ❉♦✐s ❢❡✐①❡s ❞❡ ❜♦❛✱ três ❞❡ r❡❣✉❧❛r ❡ ✉♠ ❞❡ ♠á q✉❛❧✐❞❛❞❡ sã♦ ✈❡♥❞✐❞♦s ♣♦r ✸✹ ❞♦✉✳ ❯♠ ❢❡✐①❡ ❞❡ ❜♦❛✱ ❞♦✐s ❞❡ r❡❣✉❧❛r ❡ três ❞❡ ♠á sã♦ ✈❡♥❞✐❞♦s ♣♦r ✷✻ ❞♦✉✳ ◗✉❛❧ ♦ ♣r❡ç♦ ❞♦ ❢❡✐①❡ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s q✉❛❧✐❞❛❞❡s❄✑

P❡r❝❡❜❡✲s❡ q✉❡ ♦s ❝❤✐♥❡s❡s ♦r❣❛♥✐③❛✈❛♠ ♦s ❞❛❞♦s✱ ♥❡st❡ ❝❛s♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ❡q✉❛✲ çõ❡s ❡ t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❡♠ ❢♦r♠❛ ♠❛tr✐❝✐❛❧✳ ❖s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛ ❡r❛♠ ♦r❣❛♥✐③❛❞♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

✶ ✷ ✸

✷ ✸ ✷

✸ ✶ ✶

✷✻ ✸✹ ✸✾

❚❛❜❡❧❛ ✶✳✶✿ ■♥❢♦r♠❛çõ❡s ❞♦ ♣r♦❜❧❡♠❛

◆♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ❳❱■■✱ ✉♠ ♠❛t❡♠át✐❝♦ ❥❛♣♦♥ês✱ ❚❛❦❛❦❛③✉ ❙❡❦✐ ❑♦✇❛ ✭✶✻✹✷✲✶✼✵✽✮✱ ❡st✉❞♦✉ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡✳ ❑♦✇❛ ❡r❛ ✉♠ ❞♦s ♠❛t❡♠át✐❝♦s ❞❛ ✈❛♥❣✉❛r❞❛ q✉❡ ✜③❡r❛♠ ❣r❛♥❞❡s ❞❡s❝♦❜❡rt❛s ❞❛ ♠❛t❡♠át✐❝❛ ♦❝✐❞❡♥t❛❧✳ ❈❡r❝❛ ❞❡ ✶✵ ❛♥♦s ❞❡♣♦✐s✱ ♦ ❛❧❡♠ã♦ ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ❱♦♥ ▲❡✐❜♥✐③ ✭✶✻✹✻✲✶✼✶✻✮✱ ❞❡ ❢♦r♠❛ ✐♥❞❡♣❡♥❞❡♥t❡✱ t❛♠❜é♠ ♦❜t❡✈❡ ✉♠ ♠ét♦❞♦ ❜❡♠ s✐♠✐❧❛r ❛♦ ❞❡ ❑♦✇❛ ❬✺❪✳

▲❡✐❜♥✐③ ❞❡✐①❛ ❞❡ ❢♦r♠❛ ❡①♣❧í❝✐t❛ ♥✉♠❛ ❝❛rt❛ ❡s❝r✐t❛ ♣♦r ❡❧❡ ❞❡st✐♥❛❞❛ ❛ ●✉✐❧❧❛✉♠❡ ❋r❛♥ç♦✐s ❆♥t♦✐♥❡ ▼❛rq✉✐s ❞❡ ▲✬❍ô♣✐t❛❧ ✭✶✻✻✶✲✶✼✵✹✮ ✉♠ ♠ét♦❞♦ ♣❛r❛ ❡❧✐♠✐♥❛r ✈❛❧♦r❡s ❞❡s❝♦♥❤❡❝✐❞♦s ❞❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r✳ ◆❛ ❝❛rt❛✱ ▲❡✐❜♥✐③ ❞✐③✐❛ q✉❡ ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s

10 + 11x+ 12y= 0 20 + 21x+ 22y= 0 30 + 31x+ 32y= 0

♥ã♦ ♣♦ss✉✐❛ s♦❧✉çã♦✱ ✉♠❛ ✈❡③ q✉❡ ❛ s❡❣✉✐♥t❡ ✐❣✉❛❧❞❛❞❡ 10.21.32 + 11.22.30 + 12.20.31 = 10.22.31 + 11.20.32 + 12.21.30❡r❛ ✈❡r❞❛❞❡✐r❛ ❬✻❪ ✳

❆ ❝♦♥❞✐çã♦ ❛♣r❡s❡♥t❛❞❛ ♣♦r ▲❡✐❜♥✐③ é ❡①❛t❛♠❡♥t❡ ❛ q✉❡ ❢❛③ ❝♦♠ q✉❡ ❛ ♠❛tr✐③ ❞♦s ❝♦❡✜❝✐❡♥t❡s t❡♥❤❛ ❞❡t❡r♠✐♥❛♥t❡ ♥✉❧♦✳ ❖ ❝✉r✐♦s♦ é q✉❡ ▲❡✐❜♥✐③ ♥ã♦ ✉s♦✉ ♦s ❝♦❡✜❝✐❡♥t❡s ♥✉♠ér✐❝♦s ❞❛s ❡q✉❛çõ❡s✱ ♠❛s ❞♦✐s ❝❛r❛❝t❡r❡s✱ ♦ ♣r✐♠❡✐r♦ q✉❡ ✐♥❞✐❝❛ ❛ ❡q✉❛çã♦ ♦❝♦rr❡ ❡ ♦ s❡❣✉♥❞♦ ❛ q✉❡ ✐♥❝ó❣♥✐t❛ ♣❡rt❡♥❝❡✳ P♦r ❡①❡♠♣❧♦✱ ❛q✉✐21 ❞❡♥♦t❛ ♦ q✉❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

❝♦♠♦ a21✳ ❊❧❡ ❡st❛✈❛ ❝♦♥✈❡♥❝✐❞♦ ❞❡ q✉❡ ✉♠❛ ❜♦❛ ♥♦t❛çã♦ ♠❛t❡♠át✐❝❛ ❡r❛ ❛ ❝❤❛✈❡ ♣❛r❛

♦ ♣r♦❣r❡ss♦ q✉❡ ❡❧❡ ❡①♣❡r✐♠❡♥t♦✉ ❝♦♠ ♥♦t❛çã♦ ❞✐❢❡r❡♥t❡ ♣❛r❛ ♦s s✐st❡♠❛s ❞❡ ❝♦❡✜❝✐❡♥t❡s✳ ❙❡✉s ♠❛♥✉s❝r✐t♦s ♣✉❜❧✐❝❛❞♦s ❝♦♥tê♠ ♠❛✐s ❞❡ ✺✵ ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ❞❡ r❡♣r❡s❡♥t❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❞♦ s✐st❡♠❛✳

❖ ♠❛t❡♠át✐❝♦ ❏♦❤❛♥♥ ❈❛r❧ ❋r✐❡❞r✐❝❤ ●❛✉ss ✭✶✼✼✼✲✶✽✺✺✮ ❡♠ ✶✽✵✶ ❞❡✉ s❡q✉ê♥❝✐❛ ❛♦ ❡st✉❞♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡✳ ❋♦✐ ♥❛ ♦❜r❛ ❉✐sq✉✐s✐t✐♦♥❡s ❆r✐t❤♠❡t✐❝❛❡ ✭✶✽✵✶✮ q✉❡ ●❛✉ss ✐♥tr♦❞✉③✐✉ ♦ ❝♦♥❝❡✐t♦ ✏❞❡t❡r♠✐♥❛♥t❡✑ ❛♦ ❞✐s❝✉t✐r ❢♦r♠❛s q✉❛❞rát✐❝❛s ♣♦✐s ❡ss❡ ❡❧❡♠❡♥t♦ é ♦ q✉❡ ❞❡t❡r♠✐♥❛✈❛ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❢♦r♠❛ q✉❛❞rát✐❝❛✳ ●❛✉ss ♦r❣❛♥✐③❛ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❢♦r♠❛s q✉❛❞rát✐❝❛s ♣♦r ♠❡✐♦ ❞❡ ♠❛tr✐③ r❡tâ♥❣✉❧❛r ❬✻❪✳

❏❛❝♦❜✐ ❡♠ ✶✽✹✶ ♣✉❜❧✐❝♦✉ três tr❛❜❛❧❤♦s s♦❜r❡ ❞❡t❡r♠✐♥❛♥t❡✱ s❡♥❞♦ q✉❡ ✉♠ ❞❡❧❡s tr❛③✐❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ♠❛♥❡✐r❛ ❛❧❣♦rít♠✐❝❛✱ ❡ ❛✐♥❞❛ ❛s ❡♥tr❛❞❛s ♥ã♦ ❡r❛♠ ❡s♣❡❝✐✜❝❛❞❛s ❝♦♠♦ ♥✉♠ér✐❝❛s ♦✉ ❢✉♥çõ❡s✱ ♣♦❞❡♥❞♦ s❡r q✉❛❧q✉❡r ✉♠ ❞❡st❡s ❝♦♥❝❡✐t♦s✳ ❆rt❤✉r ❈❛②❧❡② ✭✶✽✷✶✲✶✽✾✺✮ ❡♠ ✶✽✹✶ ♣✉❜❧✐❝♦✉ ❛ ♣r✐♠❡✐r❛ ❝♦♥tr✐❜✉✐çã♦ ✐♥❣❧❡s❛ ♣❛r❛ ❛ t❡♦r✐❛ ❞♦s ❞❡t❡r♠✐♥❛♥t❡✳ ◆❡st❛ ♣✉❜❧✐❝❛çã♦ ❢♦✐ ✉t✐❧✐③❛❞♦ ❞✉❛s ❧✐♥❤❛s ✈❡rt✐❝❛✐s ❡♠ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ♠❛tr✐③ ♣❛r❛ ❞❡♥♦t❛r ❞❡t❡r♠✐♥❛♥t❡✱ ✉♠❛ ❞❛s ♥♦t❛çõ❡s ❛t✉❛✐s✳ ◆❡ss❡s três tr❛❜❛❧❤♦s ❞❡ ✶✽✹✶ ❥á ❡r❛ ♣♦ssí✈❡❧ ✐❞❡♥t✐✜❝❛r q✉❡ ❛s t❛❜❡❧❛s tr❛t❛❞❛s ❛té ❡♥tã♦ ❝♦♠♦ ❝❛✐①❛s ❞❡ ❛r♠❛③❡♥❛r ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡q✉❛çõ❡s ❞❡ ✉♠ ❙✐st❡♠❛ ▲✐♥❡❛r ♥ã♦ ❡r❛♠ ❡stát✐❝❛s ❡ s✐♠ ❞✐♥â♠✐❝❛s✱ t❡♥❞♦

❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡r❡♠ ❡st✉❞❛❞❛s ❞❡ ❢♦r♠❛ s✐st❡♠❛t✐③❛❞❛✳ ❋❡r❞✐♥❛♥❞ ●♦tt❤♦❧❞ ▼❛① ❊✐s❡♥t❡✐♥ ✭✶✽✷✸✲✶✽✺✷✮ ❡♠ ✶✽✹✹✱ ❡st✉❞♦✉ ❡ss❛s t❛❜❡❧❛s ❡ ❛s ❝❤❛♠♦✉ ❞❡ ✏♠❛tr✐③❡s✑✱ ♠❡❞✐❛♥t❡ ♦ ✉s♦ ❞❡ ❧❡tr❛s ❡ s✉❛s ♦♣❡r❛çõ❡s ❝♦♠♦ ❛s r❡s✉❧t❛♥t❡s ❞❡ r❡❣r❛s ♦♣❡r❛tór✐❛s ✉s✉❛✐s s♦❜r❡ ❡ss❛s ❧❡tr❛s✱ ❝♦♠ ❡①❝❡çã♦ ❞❛ ❧❡✐ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❬✻❪✳

✶✳✷ ❖ ❡♥s✐♥♦ ❞❡ ▼❛tr✐③❡s ❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ♥❛ ❊❞✉✲

❝❛çã♦ ❇ás✐❝❛

❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ▲❡✐ ❞❡ ❉✐r❡tr✐③❡s ❡ ❇❛s❡s ❞❛ ❊❞✉❛çã♦ ◆❛❝✐♦♥❛❧ ✭▲❉❇ ✲ ▲❡✐ ✾✳✸✾✹✴✾✻✮ ❬✼❪ ❛ ❡❞✉❝❛çã♦ ❡s❝♦❧❛r ❡stá ❞✐✈✐❞✐❞❛ ❡♠ ❞✉❛s ❢❛s❡s✱ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ❡ ❡❞✉❝❛çã♦ s✉♣❡r✐♦r✳ ❙❡♥❞♦ q✉❡ ❛✐♥❞❛ ❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ é ❢♦r♠❛❞❛ ♣♦r✿ ❡❞✉❝❛çã♦ ✐♥❢❛♥t✐❧✱ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ❡♥s✐♥♦ ♠é❞✐♦✳ ❈♦♥❢♦r♠❡ ❛ ▲❉❇✱ ❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ t❡♠ ♣♦r ✜♥❛❧✐❞❛❞❡s ❞❡s❡♥✈♦❧✈❡r ♦ ❡❞✉❝❛♥❞♦✱ ❛ss❡❣✉r❛r✲❧❤❡ ❛ ❢♦r♠❛çã♦ ❝♦♠✉♠ ✐♥❞✐s♣❡♥sá✈❡❧ ♣❛r❛ ♦ ❡①❡r❝í❝✐♦ ❞❛ ❝✐❞❛❞❛♥✐❛ ❡ ❢♦r♥❡❝❡r✲❧❤❡ ♠❡✐♦s ♣❛r❛ ♣r♦❣r❡❞✐r ♥♦ tr❛❜❛❧❤♦ ❡ ❡♠ ❡st✉❞♦s ♣♦st❡r✐♦r❡s✳

❆ ♠❛t❡♠át✐❝❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ❞❡✈❡ s❡❣✉✐r ❛s ❖r✐❡♥t❛çõ❡s ❈✉rr✐❝✉❧❛r❡s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ❬✽❪✱ ❞♦❝✉♠❡♥t♦ ❡st❡ ❢♦r♠✉❧❛❞♦ ♣❡❧♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ P♦❧ít✐❝❛s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ✈✐♥❝✉❧❛❞♦ ❛♦ ▼✐♥✐stér✐♦ ❞❛ ❊❞✉❛çã♦ ✲ ▼❊❈✱ ♥♦ q✉❛❧ s❡ ❡♥❝♦♥tr❛ ♦r✐❡♥t❛çõ❡s ❞❡ ❝♦♠♦ s❡ ❞❡✈❡ tr❛❜❛❧❤❛r ♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ❞✉r❛♥t❡ ❛ ❢❛s❡ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❞❡ ♠♦❞♦ ❛ ❛t❡♥❞❡r ❛s ❝♦♠♣❡tê♥❝✐❛s ❡s♣❡❝✐✜❝❛❞❛s✳ ❖ í♥✐❝✐♦ ❞❡st❡ ❞♦❝✉♠❡♥t♦ ❛✜r♠❛ q✉❡ ♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ é ❛ss✉♠✐r q✉❡ t♦❞❛ s✐t✉❛çã♦ ❞❡ ❡♥s✐♥♦ ❛♣r❡♥❞✐③❛❣❡♠ ❞❡✈❡ ❛❣r❡❣❛r ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❤❛❜✐❧✐❞❛❞❡s q✉❡ ❝❛r❛❝t❡r✐③❡♠ ♦ ✏♣❡♥s❛r ♠❛t❡♠❛t✐❝❛♠❡♥t❡✑✳ ❆ss✐♠✱ é ♥❡❝❡ssár✐♦ ❞❛r ♣r✐♦r✐✲ ❞❛❞❡ à q✉❛❧✐❞❛❞❡ ❞♦ ♣r♦❝❡ss♦ ❡ ♥ã♦ à q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♥t❡ú❞♦s tr❛❜❛❧❤❛❞♦s✳ ❆ ❡s❝♦❧❤❛ ❞♦s ❝♦♥t❡ú❞♦s ❞❡✈❡ s❡r ❝✉✐❞❛❞♦s❛ ❡ ❝r✐t❡r✐♦s❛✱ ♣r♦♣✐❝✐❛♥❞♦ ❛♦ ❛❧✉♥♦ ✉♠ ✏❢❛③❡r ♠❛t❡♠át✐❝♦✑ ♣♦r ♠❡✐♦ ❞❡ ✉♠ ♣r♦❝❡ss♦ ✐♥✈❡st✐❣❛t✐✈♦ q✉❡ ♦ ❛✉①✐❧✐❡ ♥❛ ❛♣r♦♣r✐❛çã♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦✳ ❖s ❝♦♥t❡ú❞♦s ❜ás✐❝♦s ❡stã♦ ♦r❣❛♥✐③❛❞♦s ❡♠ q✉❛tr♦ ❜❧♦❝♦s✿ ◆ú♠❡r♦s ❡ ♦♣❡r❛çõ❡s❀ ❋✉♥çõ❡s❀ ●❡♦♠❡tr✐❛❀ ❆♥á❧✐s❡ ❞❡ ❞❛❞♦s ❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ ■ss♦ ♥ã♦ s✐❣♥✐✜❝❛ q✉❡ ♦s ❝♦♥t❡ú❞♦s ❞❡ss❡s ❜❧♦❝♦s ❞❡✈❛♠ s❡r tr❛❜❛❧❤❛❞♦s ❞❡ ❢♦r♠❛ ✐s♦❧❛❞❛s✱ ❡ s✐♠ ❞❡✈❡✲s❡ ❜✉s❝❛r ❝♦♥st❛♥t❡♠❡♥t❡ ❛ ❛rt✐❝✉❧❛çã♦ ❡♥tr❡ ❡❧❡s ❬✽❪✳ ❉❡♥tr♦ ❞❡st❡s ❜❧♦❝♦s ❡stã♦ ❛✐♥❞❛ ❛s ❝♦♠♣❡tê♥❝✐❛s q✉❡ ❞❡✈❡♠ s❡r ❞❡s❡♥✈♦❧✈✐❞❛s ❞✉r❛♥t❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠✳

P♦st❡r✐♦r♠❡♥t❡ ❢♦✐ ❡s❝r✐t❛ ❛ ▼❛tr✐③ ❞❡ ❘❡❢❡rê♥❝✐❛ ❞❡ ▼❛t❡♠át✐❝❛ ❡ s✉❛s ❚❡❝♥♦❧♦✲ ❣✐❛s ❬✾❪✱ s❡♥❞♦ ✉♠❛ ❞❛s q✉❛tr♦ ár❡❛s ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❛❜♦r❞❛❞❛ ♥♦ ❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✲ ❊◆❊▼ q✉❡ t❡♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❛✈❛❧✐❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❞♦ ❡st✉❞❛♥t❡ ❛♦ ✜♠ ❞❛ ❡s❝♦❧❛r✐❞❛❞❡ ❜ás✐❝❛ ♦♥❞❡ ♦s ♣❛rt✐❝✐♣❛♥t❡s sã♦ ❛❧✉♥♦s q✉❡ ❡stã♦ ❝♦♥❝❧✉✐♥❞♦ ♦✉ q✉❡ ❥á ❝♦♥❝❧✉ír❛♠ ♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡♠ ❛♥♦s ❛♥t❡r✐♦r❡s ❡ q✉❡ t❛♠❜é♠ é ✉t✐❧✐③❛❞♦ ❝♦♠♦ ❢♦r♠❛ ❞❡ ✐♥❣r❡ss♦ ❡♠ ■♥st✐t✉✐çõ❡s ❋❡❞❡r❛✐s ❞❡ ❊♥s✐♥♦ ❙✉♣❡r✐♦r ✲ ■❋❊❙ ♦✉ Pr✐✈❛❞❛s ♣♦r ♠❡✐♦ ❞❡

♣r♦❣r❛♠❛s ♦❢❡r❡❝✐❞♦s ♣❡❧♦ ●♦✈❡r♥♦ ❋❡❞❡r❛❧ ❝♦♠♦ ♦ Pr♦❣r❛♠❛ ❯♥✐✈❡rs✐❞❛❞❡ ♣❛r❛ ❚♦❞♦s ✲ Pr♦❯♥✐✳

P❛r❛ ❛ ❢♦r♠✉❧❛çã♦ ❞❛s ▼❛tr✐③❡s ❞❡ ❘❡❢❡rê♥❝✐❛ sã♦ ✉t✐❧✐③❛❞♦s ♦s P❛râ♠❡tr♦s ❈✉rr✐✲ ❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ✭P❈◆❊▼s✮ q✉❡ ❢♦r❛♠ ❡❧❛❜♦r❛❞♦s ♣❛r❛ ❞✐❢✉♥❞✐r ♦s ♣r✐♥❝í♣✐♦s ❞❡ r❡❢♦r♠❛ ❝✉rr✐❝✉❧❛r ❡ ♦r✐❡♥t❛r ♦s ♣r♦❢❡ss♦r❡s ❞❡ ♠❛t❡♠át✐❝❛ ♥❛ ❜✉s❝❛ ❞❡ ♥♦✈❛s ❛❜♦r❞❛❣❡♥s ❡ ♠❡t♦❞♦❧♦❣✐❛s✳

❈♦♥s✐❞❡r❛♥❞♦ ♦ tr❛t❛♠❡♥t♦ ❞❛ á❧❣❡❜r❛✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❣❡♦♠❡tr✐❛✱ ❞❛❞❛ ♣♦r ❡ss❡ ❞♦❝✉♠❡♥t♦ ♦ ❡st✉❞❛♥t❡ ❞❡✈❡ s❡r ❝❛♣❛③ ❞❡✿ ▼♦❞❡❧❛r ❡ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✲ ✈❡♠ ✈❛r✐á✈❡✐s s♦❝✐♦❡❝♦♥ô♠✐❝❛s ♦✉ té❝♥✐❝♦✲❝✐❡♥tí✜❝❛s✱ ✉s❛♥❞♦ r❡♣r❡s❡♥t❛çõ❡s ❛❧❣é❜r✐❝❛s✳ ❊ ❛✐♥❞❛ ❞❡s❡♥✈♦❧✈❡r ❛s s❡❣✉✐♥t❡s ❤❛❜✐❧✐❞❛❞❡s✿ ❘❡s♦❧✈❡r s✐t✉❛çã♦✲♣r♦❜❧❡♠❛ ❝✉❥❛ ♠♦❞❡❧❛❣❡♠ ❡♥✈♦❧✈❛ ❝♦♥❤❡❝✐♠❡♥t♦s ❛❧❣é❜r✐❝♦s✱ ✉t✐❧✐③❛r ❝♦♥❤❡❝✐♠❡♥t♦s ❛❧❣é❜r✐❝♦s✴❣❡♦♠étr✐❝♦s ❝♦♠♦ r❡✲ ❝✉rs♦ ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡ ❛r❣✉♠❡♥t❛çã♦ ❡ ❆✈❛❧✐❛r ♣r♦♣♦st❛s ❞❡ ✐♥t❡r✈❡♥çã♦ ♥❛ r❡❛❧✐❞❛❞❡ ✉t✐❧✐③❛♥❞♦ ❝♦♥❤❡❝✐♠❡♥t♦s ❛❧❣é❜r✐❝♦s✳

P❛r❛ s❡ ❞❡s❡♥✈♦❧✈❡r ❡ss❛s ❤❛❜✐❧✐❞❛❞❡s✱ ♥♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s✱ ❛❧é♠ ❞❡ tr❛❜❛❧❤❛r ❛ té❝♥✐❝❛ ❞❡ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s✱ é r❡❝♦♠❡♥❞á✈❡❧ ❢❛③❡r ❛ ❧✐❣❛çã♦ ❡♥tr❡ ❛ á❧❣❡❜r❛ ❡ ❛ ❣❡♦♠❡tr✐❛✱ ✈❛❧❡♥❞♦ r❡ss❛❧t❛r q✉❡ ✐ss♦ é ♣♦ssí✈❡❧ ❡♠ ❝❛s♦s ❞❡ ❡q✉❛çõ❡s ❝♦♠ ❞✉❛s ✈❛r✐á✈❡✐s✱ s❡♥❞♦ ❡st❛ r❡♣r❡s❡♥t❛❞❛ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ♣♦r ✉♠❛ r❡t❛✱ ♦✉ ❛✐♥❞❛ ♥♦ ❝❛s♦ ❞❡ ❡q✉❛çõ❡s ❝♦♠ três ✈❛r✐á✈❡✐s✱ q✉❡ ♣♦❞❡♠ s❡r ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ r❡t❛ ♦✉ ✉♠ ♣❧❛♥♦ ♥♦ ❡s♣❛ç♦✱ s❡♥❞♦ q✉❡ ❡♠ t♦❞♦s ♦s ❝❛s♦s t❡♠✲s❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❬✽❪✳

❆ r❡s♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ 2×2✱ ♦✉ s❡❥❛✱ ❞❡ ❞✉❛s ❡q✉❛çõ❡s ❡ ❞✉❛s ✈❛r✐á✈❡✐s✱ ♣♦❞❡

❡st❛r ❛ss♦❝✐❛❞❛ ❛♦ ❡st✉❞♦ ❞❛ ♣♦s✐çã♦ r❡❧❛t✐✈❛ ❞❡ ❞✉❛s r❡t❛s ♥♦ ♣❧❛♥♦✳ P♦r ♠❡✐♦ ❞❡ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s s✐♠♣❧❡s✱ ♣♦❞❡✲s❡ ❞❡t❡r♠✐♥❛r ❛ ❡①✐stê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ s♦❧✉çõ❡s ❞❡ss❡ s✐st❡♠❛✱ ♦ q✉❡ s✐❣♥✐✜❝❛ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ♦s ❝❛s♦s ❞❡ ✐♥t❡rs❡❝çã♦✴❝♦✐♥❝✐❞ê♥❝✐❛ ❞❡ r❡t❛s ♦✉ ♣❛r❛❧❡❧✐s♠♦ ❞❡ r❡t❛s✳ ❆ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s 2×3 ♦✉ 3×3 ❞❡✈❡ s❡r ❢❡✐t❛ ♣♦r ♠❡✐♦ ❞❡ ♦♣❡r❛çõ❡s

❡❧❡♠❡♥t❛r❡s ✭♦ ♣r♦❝❡ss♦ ❞❡ ❡s❝❛❧♦♥❛♠❡♥t♦✱ q✉❡ s❡rá ❛♣r❡s❡♥t❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ✷✮✱ ❝♦♠ ❞✐s❝✉ssã♦ ❞❛s ❞✐❢❡r❡♥t❡s s✐t✉❛çõ❡s✳ ❆ r❡❣r❛ ❞❡ ❈r❛♠❡r✱ ✉s❛❞❛ ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çã♦ 3× 3✱ ❞❡✈❡r s❡r ❛❜❛♥❞♦♥❛❞❛✱ ♣♦r s❡r ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ❝✉st♦s♦ ✭❡♠ ❣❡r❛❧✱

❛♣r❡s❡♥t❛❞♦ s❡♠ ❞❡♠♦♥str❛çã♦✱ ❡✱ ♣♦rt❛♥t♦ ❞❡ ♣♦✉❝♦ s✐❣♥✐✜❝❛❞♦ ♣❛r❛ ♦ ❛❧✉♥♦✮✱ ❡♠ q✉❡ é ♣♦ssí✈❡❧ ❛♣❧✐❝❛çã♦ ❡♠ s✐st❡♠❛s ❛♣❡♥❛s ❝♦♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❡q✉❛çõ❡s ❡ ✈❛r✐á✈❡✐s✱ ❝❤❛♠❛❞♦s ❞❡ q✉❛❞r❛❞♦s✳ ❉❡t❡r♠✐♥❛♥t❡ ❞❡✈❡ s❡r ❞✐s♣❡♥s❛❞♦✱ ✉♠❛ ✈❡③ q✉❡ ❛ ✉t✐❧✐③❛çã♦ s❡ ❞á ❛♣❡♥❛s ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❝♦♠ ❛ ❘❡❣r❛ ❞❡ ❈r❛♠❡r ❬✽❪✳

✶✳✸ ❖❜❥❡t✐✈♦s

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❞❡ ❝♦♥❝❡✐t♦s ❝♦♠♦ ▼❛tr✐③❡s✱ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❡ ❉❡t❡r♠✐♥❛♥t❡ ✈✐st♦s ❞✉r❛♥t❡ ♦ ✷➸ ❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡ ♥❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r ❡♠ ♥í✈❡❧ ❞❡ ❣r❛❞✉❛çã♦ ❡♠ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡♠ ♣r♦❜❧❡♠❛s r❡❛✐s ❞❛ ❊♥❣❡♥❤❛r✐❛ ❈✐✈✐❧✱ t❛✐s ❝♦♠♦✿ r❡s♦❧✉çõ❡s ❞❡ tr❡❧✐ç❛s ❡ ♥❛s ❞❡t❡r♠✐♥❛çõ❡s ❞♦s ♠♦❞♦s ❞❡ ✈✐❜r❛çã♦ ❞❡ ✉♠❛ ❡str✉t✉r❛✳ P❛r❛ ✐ss♦ s❡rá ♥❡❝❡ssár✐♦ ❞❡s❡♥✈♦❧✈❡r ❛ ♣❛rt❡ t❡ór✐❝❛ ❞♦s tó♣✐❝♦s s✉♣r❛❝✐t❛❞♦s ❡ ❛♣r❡s❡♥t❛r ❝♦♥❝❡✐t♦s ❢ís✐❝♦s ✐♠♣r❡s❝✐♥❞í✈❡✐s ♣❛r❛ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❞❛s ❛♣❧✐❝❛çõ❡s q✉❡ s❡rã♦ ❢❡✐t❛s✳

❉❡ss❛ ❢♦r♠❛✱ ❛ ♦r❣❛♥✐③❛çã♦ ❞♦ tr❛❜❛❧❤♦ s❡rá ❢❡✐t❛ ❝♦♠♦ s❡❣✉❡✿

❈❛♣ít✉❧♦ ✷✿ ❙❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ ❞♦s ❝♦♥❝❡✐t♦s ▼❛tr✐③❡s ❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s✱ ❛ ✐♥❝❧✉✐r✱ ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ t❡♦r❡♠❛s✱ ❝✉❥♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ é ♠♦str❛r ❛ ❧✐❣❛çã♦ ❡①✐st❡♥t❡ ❡♥tr❡ ♦s ❞♦✐s ❛ss✉♥t♦s✳ P❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s s❡rã♦ ✉t✐❧✐③❛❞♦s ▼ét♦❞♦s ❉✐r❡t♦s ❡ ■t❡r❛t✐✈♦s✱ s❡♥❞♦ ❡st❡ ú❧t✐♠♦ ❝♦♠ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❡♠ ♣r♦❜❧❡♠❛s ❝♦♥❝r❡t♦s✱ ❞❡✈✐❞♦ ❛♦ ❡❧❡✈❛❞♦ ♥ú♠❡r♦ ❞❡ ❡q✉❛çõ❡s✳

❈❛♣ít✉❧♦ ✸✿ ➱ ❡①✐❜✐❞♦ ❛ ❞❡✜♥✐çã♦ ❢♦r♠❛❧ ❞❡ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛✳ P❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❛♥❛❧ít✐❝♦ ❞❡ ❉❡t❡r♠✐♥❛♥t❡✱ é ✐♠♣♦rt❛♥t❡ ♦ ❡st✉❞♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♠♦❞♦ ❛ ❢❛❝✐❧✐t❛r ❛ s✉❛ ♦❜t❡♥çã♦✳ ❙❡rã♦ ❛♣r❡s❡♥t❛❞♦s ♠ét♦❞♦s ❞❡ r❡s♦❧✉çã♦ ❞❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ♣♦r ♠❡✐♦ ❞❡ ❉❡t❡r♠✐♥❛♥t❡ ❡ ♠ét♦❞♦s ❞❡ ♦❜t❡♥çã♦ ❞❛ ♠❛tr✐③ ✐♥✈❡rs❛✳

❈❛♣ít✉❧♦ ✹✿ ❙❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❞✉❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❡ ❉❡t❡r♠✐✲ ♥❛♥t❡✱ ♣r✐♠❡✐r♦ ♥❛ r❡s♦❧✉çã♦ ❞❡ tr❡❧✐ç❛s ❡ s❡❣✉♥❞♦ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦s ♠♦❞♦s ❞❡ ✈✐❜r❛çã♦ ❞❡ ✉♠❛ ❡str✉t✉r❛✳ ❆ ♠❡t♦❞♦❧♦❣✐❛ ✉s❛❞❛ ✈✐s❛ ❛ ❛♣❧✐❝❛çã♦ ❞♦s ❝♦♥❝❡✐t♦s ❛❜♦r❞❛❞♦s ❞✉r❛♥t❡ t♦❞❛ ❞✐ss❡rt❛çã♦✳ P❛r❛ ❝♦♠♣♦r ❡ss❛s ❛♣❧✐❝❛çõ❡s s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❝♦♥❝❡✐t♦s ♥ã♦ ✈✐st♦s ❞✉r❛♥t❡ ♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❝♦♠♦ tr❡❧✐ç❛s ❡ ✈✐❜r❛çõ❡s✳

❊ ♣♦r ✜♠✱ ♥♦ ❈❛♣ít✉❧♦ ✺✱ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛s ❈♦♥❝❧✉sõ❡s ❞♦ ❚r❛❜❛❧❤♦✳

❈❛♣ít✉❧♦ ✷

▼❛tr✐③❡s ❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦ s❡rá ✈✐st♦ ❞❡ ❢♦r♠❛ s✐st❡♠❛t✐③❛❞❛ t♦❞❛s ❛s ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ t❡♦r❡♠❛s ♣❛r❛ ✉♠❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ ❛❝❡r❝❛ ❞❡ ▼❛tr✐③❡s ❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s✳ ❆s ❢♦r♠❛s ❞❡ r❡♣r❡s❡♥t❛çã♦ ♠❛tr✐❝✐❛❧ ❛♣❛r❡❝❡♠ ❞❡ ❢♦r♠❛ ♥❛t✉r❛❧ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♠✉✐t♦s ♣r♦❜❧❡♠❛s✱ ♥ã♦ ❛♣❡♥❛s ♣♦rq✉❡ ❡❧❡s ✏♦r❞❡♥❛♠ ❡ s✐♠♣❧✐✜❝❛♠✑ ❛ ❧✐♥❣✉❛❣❡♠ ❞♦ ♣r♦❜❧❡♠❛✱ ♠❛s t❛♠❜é♠ ♣♦rq✉❡ ♣♦ss✐❜✐❧✐t❛♠ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ♥♦✈♦s ♠ét♦❞♦s ❞❡ r❡s♦❧✉çã♦ ❛tr❛✈és ❞❡ ❢❡rr❛♠❡♥t❛s ❝♦♠♦ ♦ ❝♦♠♣✉t❛❞♦r✱ s❡♥❞♦ ❡st❡ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♥❛ r❡s♦❧✉çã♦ ❞❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s ❝♦♠ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ✈❛r✐á✈❡✐s✳ ❙❡rã♦ ❛♣r❡s❡♥t❛❞♦s ▼ét♦❞♦s ❉✐r❡t♦s ✭❊s❝❛❧♦♥❛♠❡♥t♦ ❡ ❈r❛♠❡r✮ ❡ ■t❡r❛t✐✈♦s ✭Pr♦❝❡ss♦ ❞❡ ❏❛❝♦❜✐✲❘✐❝❤❛r❞s♦♥ ❡ Pr♦❝❡ss♦ ❞❡ ●❛✉ss✲❙❡✐❞❡❧✮ ❞❡ r❡s♦❧✉çã♦ ❞❡ ❙✐st❡♠❛s ▲✐♥❡❛r❡s✳

✷✳✶ ❉❡✜♥✐çõ❡s ❇ás✐❝❛s ❡ ❚✐♣♦s ❞❡ ▼❛tr✐③❡s

❉❡✜♥✐çã♦ ✷✳✶✳ ▼❛tr✐③

❙❡❥❛♠ m ≥ 1 ❡ n ≥ 1 ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❯♠❛ ♠❛tr✐③ m ×n r❡❛❧ é ❛ ❞✉♣❧❛

s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ ❞✐str✐❜✉í❞♦s ❡♠ ♠ ❧✐♥❤❛s ❡ ♥ ❝♦❧✉♥❛s✱ ❢♦r♠❛♥❞♦ ✉♠❛ t❛❜❡❧❛ q✉❡ s❡ ✐♥❞✐❝❛ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿

A=

     

a11 a12 ... a1n

a21 a22 ... a2n ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

am1 am2 ... amn

     

❆❜r❡✈✐❛❞❛♠❡♥t❡ ❡st❛ ♠❛tr✐③ ♣♦❞❡ s❡r ❡①♣r❡ss❛ ♣♦r A = (aij), ❝♦♠ i = 1,2, .., m ❡ j = 1,2, ..., n✳ ❈❛❞❛ ❡❧❡♠❡♥t♦ q✉❡ ❝♦♠♣õ❡ ✉♠❛ ♠❛tr✐③ ❝❤❛♠❛✲s❡ t❡r♠♦ ❞❡ss❛ ♠❛tr✐③✳

❉❛❞❛ ❛ ♠❛tr✐③ A = (aij)✱ ❛♦ sí♠❜♦❧♦ aij q✉❡ r❡♣r❡s❡♥t❛ ✐♥❞✐st✐♥t❛♠❡♥t❡ t♦❞♦s ♦s s❡✉s t❡r♠♦s ❞á✲s❡ ♦ ♥♦♠❡ ❞❡ t❡r♠♦ ❣❡r❛❧ s❡♥❞♦ q✉❡ ♦ í♥❞✐❝❡i✐♥❞✐❝❛ ❛ ❧✐♥❤❛ ❛♦ q✉❛❧ ♦ ❡❧❡♠❡♥t♦

♣❡rt❡♥❝❡ ❡ ♦ í♥❞✐❝❡j ❛ ❝♦❧✉♥❛ ❛♦ q✉❛❧ ♦ ❡❧❡♠❡♥t♦ ♣❡rt❡♥❝❡ ❝♦♠ ❛s ❧✐♥❤❛s ♥✉♠❡r❛❞❛s ❞❡1

❛té m✱ ❡ ❛s ❝♦❧✉♥❛s ♥✉♠❡r❛❞❛s ❞❡ 1 ❛té n✳ ❙❡rá ✐♥❞✐❝❛❞♦ ♣♦r Mm×n(R) ♦ ❝♦♥❥✉♥t♦ ❞❛s ♠❛tr✐③❡s r❡❛✐sm×n✱ s❡♥❞♦ q✉❡ é ✉s✉❛❧ r❡♣r❡s❡♥t❛r ✉♠❛ ♠❛tr✐③ ❞❡Mm×n(R)s✐♠♣❧❡s♠❡♥t❡ ♣♦r Am×n✳ ❙❡ m = n✱ s❡rá ✉s❛❞❛ ❛ ♥♦t❛çã♦ Mn(R)✱ ❡ ❝❛❞❛ ♠❛tr✐③ ❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ q✉❛❞r❛❞❛ ❞❡ ♦r❞❡♠ ♥ ❬✶✵❪✳

▼❛tr✐③❡s ❊s♣❡❝✐❛✐s

❛✮matriz linha✿ ❚♦❞❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠1×n✱ ✐st♦ é✱ ✉♠❛ ♠❛tr✐③ q✉❡ ♣♦ss✉✐ ❛♣❡♥❛s

✉♠❛ ❧✐♥❤❛✳

M =h a11 a12 ... a1n

i

1×n

❜✮ matriz coluna✿ ❚♦❞❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠ m ×1✱ ✐st♦ é✱ ✉♠❛ ♠❛tr✐③ q✉❡ ♣♦ss✉✐

❛♣❡♥❛s ✉♠❛ ❝♦❧✉♥❛✳

M =       a11 a21 ✳✳✳

am1

      m×1

❝✮ matriz nula✿ ❚♦❞❛ ♠❛tr✐③✱ ❡♠ q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ aij✱ t❡♠✲s❡aij = 0✳

0m×n=

     

0 0 ... 0 0 0 ... 0

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

0 0 ... 0

      m×n

❞✮ matriz quadrada✿ ❚♦❞❛ ♠❛tr✐③✱ ❡♠ q✉❡✱ ♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s é ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦

❞❡ ❝♦❧✉♥❛s✳ M =      

a11 a12 ... a1n

a21 a22 ... a2n ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

an1 an2 ... ann

      n×n

❡✮matriz triangular inf erior✿ ❚♦❞❛ ♠❛tr✐③ q✉❛❞r❛❞❛✱ ♥❛ q✉❛❧✱ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s aij✱ ❡♠ q✉❡✱i < j sã♦ ✐❣✉❛✐s ❛ ③❡r♦✳

M =      

a11 0 ... 0

a21 a22 ... 0

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

an1 an2 ... ann

❢✮ matriz triangular superior✿ ❚♦❞❛ ♠❛tr✐③ q✉❛❞r❛❞❛✱ ♥❛ q✉❛❧✱ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s aij✱ ❡♠ q✉❡✱i > j sã♦ ✐❣✉❛✐s ❛ ③❡r♦✳

M =      

a11 a12 ... a1n

0 a22 ... a2n ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

0 0 ... ann

      n×n

❣✮ matriz diagonal✿ ❚♦❞❛ ♠❛tr✐③ q✉❛❞r❛❞❛✱ ♥❛ q✉❛❧✱ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s aij✱ ❡♠ q✉❡✱i6=j sã♦ ✐❣✉❛✐s ❛ ③❡r♦✳

M =      

a11 0 ... 0

0 a22 ... 0

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

0 0 ... ann

      n×n

❤✮matriz identidade✿ ❚♦❞❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧✱ ♥❛ q✉❛❧✱ t♦❞♦s ♦s ❡❧❡♥t♦saij sã♦ ✐❣✉❛✐s ❛1 ♣❛r❛ i=j ❡0 ♣❛r❛ i6=j ✳

In =

     

1 0 ... 0

0 1 ... 0

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

0 0 ... 1

      n×n

✐✮ matriz escada✿ ❯♠❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠ m×n s❡rá ❞✐t❛ ♥❛ ❢♦r♠❛ ❡s❝❛❞❛ s❡ ❢♦r

♥✉❧❛✱ ♦✉ s❡✿

(i) ❖ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ ♥ã♦ ♥✉❧♦ ❞❡ ✉♠❛ ❧✐♥❤❛ ♥ã♦ ♥✉❧❛ é ♦ ♥ú♠❡r♦1.

(ii) ❈❛❞❛ ❝♦❧✉♥❛ q✉❡ ❝♦♥té♠ ♦ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ ♥ã♦ ♥✉❧♦ ❞❡ ❛❧❣✉♠❛ ❧✐♥❤❛ t❡♠

t♦❞♦s ♦s s❡✉s ♦✉tr♦s ❡❧❡♠❡♥t♦s ✐❣✉❛✐s ❛ ③❡r♦✳

(iii) ❚♦❞❛ ❧✐♥❤❛ ♥✉❧❛ ♦❝♦rr❡ ❛❜❛✐①♦ ❞❡ t♦❞❛s ❛s ❧✐♥❤❛s ♥ã♦ ♥✉❧❛s✳

(iv)❙❡ ❛s ❧✐♥❤❛s 1,2, ..., r sã♦ ❛s ❧✐♥❤❛s ♥ã♦ ♥✉❧❛s✱ ❡ s❡ ♦ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ ♥ã♦ ♥✉❧♦

❞❛ ❧✐♥❤❛i ♦❝♦rr❡ ♥❛ ❝♦❧✉♥❛ Ki ❡♥tã♦ K1 < K2 < ... < Ki. ❊st❛ ú❧t✐♠❛ ❝♦♥❞✐çã♦ ✐♠♣õ❡ ❛ ❢♦r♠❛ ❡s❝❛❞❛ à ♠❛tr✐③ ❬✶✵❪✳

❉❡ss❛ ❢♦r♠❛✱ ✉♠❛ ♠❛tr✐③ ❡st❛rá ♥❛ ❢♦r♠❛ ❡s❝❛❞❛ s❡✱ ♦ ♥ú♠❡r♦ ❞❡ ③❡r♦s ♣r❡❝❡❞❡♥❞♦ ♦ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ ♥ã♦ ♥✉❧♦ ❞❡ ✉♠❛ ❧✐♥❤❛ ❛✉♠❡♥t❛ ❛ ❝❛❞❛ ❧✐♥❤❛✱ ❛té q✉❡ s♦❜r❡♠ s♦♠❡♥t❡ ❧✐♥❤❛s ♥✉❧❛s✱ s❡ ❤♦✉✈❡r✳

✷✳✷ ❖♣❡r❛çõ❡s ❝♦♠ ▼❛tr✐③❡s

❉❡✜♥✐çã♦ ✷✳✷✳ ■❣✉❛❧❞❛❞❡ ❞❡ ▼❛tr✐③❡s

❙❡❥❛♠ A, B∈ Mm×n(R)✳ ❚❡♠✲s❡ q✉❡ A ❡ B sã♦ ✐❣✉❛✐s✱ ❡ ❡s❝r❡✈❡✲s❡ A = B s❡✱ ❡ s♦♠❡♥t❡ s❡✱ aij =bij✱ ♣❛r❛ t♦❞♦ i= 1, . . . , m❡ t♦❞♦ j = 1, . . . , n✳

❉❡✜♥✐çã♦ ✷✳✸✳ ❆❞✐çã♦ ❞❡ ▼❛tr✐③❡s

❙❡❥❛♠ A, B∈ Mm×n(R)✳ ■♥❞✐❝❛✲s❡ ♣♦r A+B ❡ ❝❤❛♠❛✲s❡ s♦♠❛ ❞❡ A ❝♦♠ B ❛ ♠❛tr✐③m×n ❝✉❥♦ t❡r♠♦ ❣❡r❛❧ é aij +bij✱ ♦✉ s❡❥❛

A+B =

     

a11+b11 a12+b12 ... a1n+b1n

a21+b21 a22+b22 ... a2n+b2n

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

am1+bm1 am2+bm2 ... amn+bmn

     

❉á✲s❡ ♦ ♥♦♠❡ ❞❡ ❛❞✐çã♦ à ♦♣❡r❛çã♦ q✉❡ tr❛♥s❢♦r♠❛ ❝❛❞❛ ♣❛r(A, B)❞❡ ♠❛tr✐③❡s r❡❛✐s

❞♦ ♠❡s♠♦ t✐♣♦m×n ♥❛ ♠❛tr✐③ A+B✳ ❊st❛ é ✉♠❛ ♦♣❡r❛çã♦ ♥♦ ❝♦♥❥✉♥t♦ Mm×n(R).

Pr♦♣r✐❡❞❛❞❡s

❙❡❥❛♠ A, B, C∈ Mm×n(R)✱ t❡♠✲s❡✿

i)A+B =B+A ✭❝♦♠✉t❛t✐✈✐❞❛❞❡✮❀

ii)A+ (B +C) = (A+B) +C ✭❛ss♦❝✐❛t✐✈✐❞❛❞❡✮❀

iii)A+ 0m×n=A✱ s❡♥❞♦0m×n r❡♣r❡s❡♥t❛ ❛ ♠❛tr✐③ ♥✉❧❛ ❞❡Mm×n(R)❀

iv)A+ (−A) = 0m×n✱ s❡♥❞♦(−A) é ❛ ♠❛tr✐③ ♦♣♦st❛ ❞❛ ♠❛tr✐③ A✳

❉❡♠♦♥str❛çã♦✳ ❆ ❞❡♠♦♥str❛çã♦ ❞♦s ✐t❡♥s ❞❡❝♦rr❡♠ ❞✐r❡t❛♠❡♥t❡ ❞❛s ❞❡✜♥✐çõ❡s ❞❡ ✐❣✉❛❧✲ ❞❛❞❡ ❡ ❛❞✐çã♦ ❞❡ ♠❛tr✐③❡s✳ P♦r ❡ss❡ ♠♦t✐✈♦ s❡rá ♣r♦✈❛❞♦ ❛♣❡♥❛s ♦ ✐t❡♠ (ii)✱ ❛ ❞❡♠♦♥s✲

tr❛çã♦ ❞♦s ✐t❡♥s (i), (iii) ❡(iv)✱ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✶✶❪✳

(ii)✿ ❙❡❥❛♠ A= (aij), B = (bij), C = (cij)∈ Mm×n(R)✱ ❡♥tã♦

A+(B+C) = (aij)+(bij +cij) = (aij + (bij +cij)) = ((aij +bij) +cij) = (aij +bij)+

(cij) = (A+B) +C.

❆s ♦♣❡r❛çõ❡s q✉❡ s❡rã♦ ❞❡✜♥✐❞❛s ❛ s❡❣✉✐r ✉t✐❧✐③❛♠ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠❛ ♠❛tr✐③ ♣♦r ✉♠ ♥ú♠❡r♦ r❡❛❧✱ t❛♠❜é♠ ❝❤❛♠❛❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠❛ ♠❛tr✐③ ♣♦r ❡s❝❛❧❛r✳

❉❡✜♥✐çã♦ ✷✳✹✳ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠❛ ▼❛tr✐③ ♣♦r ❡s❝❛❧❛r

❙❡❥❛ A = (aij) ∈ Mm×n(R) ❡ k ✉♠ ♥ú♠❡r♦ r❡❛❧✱ ♦ ♣r♦❞✉t♦ ❞❡ k ♣♦r A é ❛ ♠❛tr✐③ r❡❛❧✱ m×n ❝✉❥♦ t❡r♠♦ ❣❡r❛❧ é ❛ ♠❛tr✐③ k.A= (kaij)✱ ♦✉ s❡❥❛✿

k.A=

     

ka11 ka12 ... ka1n

ka21 ka22 ... ka2n ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

kam1 kam2 ... kamn

     

❉❡ss❛ ❢♦r♠❛ s❡ ❞❡✜♥✐✉ ❛ ♦♣❡r❛çã♦ q✉❡ tr❛♥s❢♦r♠❛ ❝❛❞❛ ♣❛r(k, A)❞❡R× Mm×n(R) ♥❛ ♠❛tr✐③ r❡❛❧ kA∈ Mm×n(R)✳

Pr♦♣r✐❡❞❛❞❡s

❙❡❥❛♠ A= (aij)✱ B = (bij)∈ Mm×n(R)✱ ❡k1, k2 ∈R✱ t❡♠✲s❡✿

i)k1(A+B) =k1A+k1B❀

ii) (k1+k2)A=k1A+k2A❀

iii) 0.A= 0m×n❀

iv)k1(k2A) = (k1k2)A❀

v) 1.A=A ✭❡❧❡♠❡♥t♦ ♥❡✉tr♦✮✳

❉❡♠♦♥str❛çã♦✳ (i)✿ ❉❡ ❢❛t♦✱ s❡❥❛♠ A= (aij)✱ B = (bij)∈ Mm×n(R)❡ k ∈R✱ ❡♥tã♦

k(A+B) = k(aij +bij) = (k(aij+bij)) = (kaij+kbij) = (kaij) + (kbij) = k(aij) +

k(bij) = kA+kB.

❉❡✜♥✐çã♦ ✷✳✺✳ ❚r❛♥s♣♦s✐çã♦

❙❡❥❛ A = (aij) ∈ Mm×n(R) ❞❡♥♦♠✐♥❛✲s❡ tr❛♥s♣♦st❛ ❞❡ A ❡ ✐♥❞✐❝❛✲s❡ ♣♦r AT ❛ s❡❣✉✐♥t❡ ♠❛tr✐③n×m✿ AT _{= (b}

ji)✱ ♦♥❞❡ bji =aij ❝♦♠ i= 1,2, ..., m❡ j = 1,2, ..., n✳ Pr♦♣r✐❡❞❛❞❡s

❙❡❥❛♠ A= (aij)✱B = (bij)∈ Mm×n(R) ❡k ∈R✱ t❡♠✲s❡✿

i) (AT₎T _=A❀

ii) (A+B)T _=AT _+BT_❀

iii) (kA)T _=kAT_✳

❉❡♠♦♥str❛çã♦✳ (ii) ❙❡❥❛♠ A= (aij)✱ B = (bij)∈ Mm×n(R)✳ ❊♥tã♦✱

(A+B)T _{= (a}

ji+bji) = (aji) + (bji) = AT +BT✳

❆s ❞❡♠♦♥str❛çõ❡s ❞♦s ❞❡♠❛✐s ✐t❡♥s ❛❝❡r❝❛ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r ❡ tr❛♥s♣♦s✐çã♦ é ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✶✶❪✳

❉❡✜♥✐çã♦ ✷✳✻✳ ▼❛tr✐③ ❙✐♠étr✐❝❛

❯♠❛ ♠❛tr✐③ A = (aij) ∈ Mn(R)✱ é sim´etrica s❡✱ ❡ s♦♠❡♥t❡ s❡✱ AT = A✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦aij =aji✳

❉❡✜♥✐çã♦ ✷✳✼✳ ▼❛tr✐③ ❆♥t✐ ❙✐♠étr✐❝❛

❯♠❛ ♠❛tr✐③ A= (aij)∈ Mn(R)✱ é anti sim´etrica s❡✱ ❡ s♦♠❡♥t❡ s❡✱AT =−A✱ ✐st♦ é✱ q✉❛♥❞♦aij =−aji✳

❉❡✜♥✐çã♦ ✷✳✽✳ Pr♦❞✉t♦ ❞❡ ▼❛tr✐③❡s

❙❡❥❛♠ A = (aij)∈ Mm×p(R) ❡ B = (bjk) ∈ Mp×n(R)✱ ❞❡♥♦♠✐♥❛✲s❡ ♣r♦❞✉t♦ ❞❡ A ♣♦rB ❡ r❡♣r❡s❡♥t❛✲s❡ ♣♦r AB ❛ ♠❛tr✐③C = (cik)∈ Mm×n(R)✱ ❝✉❥♦ t❡r♠♦ ❣❡r❛❧ é t❛❧ q✉❡

cik =ai1.b1k+ai2.b2k+· · ·+ain.bnk = n

X

j=1

aij.bjk,∀i∈ {1,2,· · · , m}❡ ∀k∈ {1,2,· · · , p}✳

❊st❛ ♦♣❡r❛çã♦ é ❞❡✜♥✐❞❛ ❛♣❡♥❛s q✉❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s ❞❛ ♠❛tr✐③Aé ♦ ♠❡s♠♦

♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❞❛ ♠❛tr✐③ B✳

Pr♦♣r✐❡❞❛❞❡s

❙❡❥❛♠ A✱ B ❡ C ♠❛tr✐③❡s r❡❛✐s✱ t❡♠✲s❡✿

i)AIn=InA=A✱ ❝♦♠ A∈ Mm×n(R)❀

ii)A(B+C) =AB+AC✱ ❝♦♠ A∈ Mm×p(R)❡ B, C ∈ Mp×n(R)❀

iii) (A+B)C=AC+BC✱ ❝♦♠ A, B ∈ Mm×p(R) ❡ C∈ Mp×n(R)❀

iv)A(BC) = (AB)C✱ ❝♦♠ A∈ Mm×p(R)✱B ∈ Mp×q(R) ❡C ∈ Mq×n(R)❀

v) (AB)T _=BT_AT_{✱ ❝♦♠ A∈ M}

m×p(R) ❡ B ∈ Mp×n(R)✳

◆ã♦ é ✈á❧✐❞❛ ❛ ♣r♦♣r✐❡❞❛❞❡ ❝♦♠✉t❛t✐✈❛ ♥❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s✱ ♣♦✐s ❡♠ ❣❡r❛❧✱

AB 6=BA✳

❉❡✜♥✐çã♦ ✷✳✾✳ ▼❛tr✐③ ❖rt♦❣♦♥❛❧