Antonio Carlos Marques da Silva Ana Paula Lima Marques Fernandes

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛

■♥tr♦❞✉çã♦ à ➪❧❣❡❜r❛ ▲✐♥❡❛r

❆♥t♦♥✐♦ ❈❛r❧♦s ▼❛rq✉❡s ❞❛ ❙✐❧✈❛

❈❛t❛❧♦❣❛çã♦ ♥❛ ❢♦♥t❡

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s ❇✐❜❧✐♦t❡❝❛ ❈❡♥tr❛❧

❉✐✈✐sã♦ ❞❡ ❚r❛t❛♠❡♥t♦ ❚é❝♥✐❝♦

■♥tr♦❞✉çã♦ à ➪❧❣❡❜r❛ ▲✐♥❡❛r

❙❯▼➪❘■❖

• ❆♣r❡s❡♥t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ • ❈❛♣ít✉❧♦ ✶ ✕ ❙✐st❡♠❛s ❧✐♥❡❛r❡s ❡ ♠❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ◮❘❡s♣♦st❛s ❞❛s ❆t✐✈✐❞❛❞❡s✲♣r♦♣♦st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ • ❈❛♣ít✉❧♦ ✷ ✕ ❊s♣❛ç♦s ✈❡t♦r✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ◮❘❡s♣♦st❛s ❞❛s ❆t✐✈✐❞❛❞❡s✲♣r♦♣♦st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ • ❈❛♣ít✉❧♦ ✸ ✕ ❇❛s❡s ❡ ❞✐♠❡♥sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ◮❘❡s♣♦st❛s ❞❛s ❆t✐✈✐❞❛❞❡s✲♣r♦♣♦st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ • ❈❛♣ít✉❧♦ ✹ ✕ ❚r❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ◮❘❡s♣♦st❛s ❞❛s ❆t✐✈✐❞❛❞❡s✲♣r♦♣♦st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ • ❈❛♣ít✉❧♦ ✺ ✕ ❊s♣❛ç♦s ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ◮❘❡s♣♦st❛s ❞❛s ❆t✐✈✐❞❛❞❡s✲♣r♦♣♦st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼ • ❆♥❡①♦ ✕ ❉❡t❡r♠✐♥❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ • ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾

❆P❘❊❙❊◆❚❆➬➹❖

❉❛♥❞♦ s❡q✉❡♥❝✐❛ ❛ s❡✉ ❝✉rs♦ ❞❡ ❊❆❉ ✖ ▲✐❝❡♥❝✐❛t✉r❛ ❡♠ ▼❛t❡♠át✐❝❛ ✖✱ ♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❆▲ ♥♦s ❤♦♥r♦✉ ❝♦♠ ♦ ❝♦♥✈✐t❡ ❞❡ ❡❧❛❜♦r❛r ✉♠ t❡①t♦ ✐♥tr♦❞✉tór✐♦ ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r✱ ❝♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ ❛❧✐❝❡rç❛r ❛s ✈❛r✐❛❞❛s ❛♣❧✐❝❛çõ❡s ❞❡ss❛ ❡①tr❛♦r❞✐♥ár✐❛ ❞✐s❝✐♣❧✐♥❛✳

❖ ❛✉t♦r sê♥✐♦r✱ q✉❡ ❡s❝♦❧❤❡✉ ♦ s❡t♦r ❞❡ ❡st✉❞♦s ❞❡ ➪❧❣❡❜r❛ ♣❛r❛ s✉❛s ❛t✐✈✐❞❛❞❡s ❛❝❛❞ê♠✐❝❛s ❡ ❞❡ ♣❡sq✉✐s❛s✱ ✈❡♠ ❛❝♦♠♣❛♥❤❛♥❞♦ ♦s tr❛❜❛❧❤♦s ❞❛ ❥♦✈❡♠ ♣r♦❢❡ss♦r❛ ❆♥❛ P❛✉❧❛✱ ❝✉❥❛ ❛tr❛çã♦ ♣❡❧❛s ❞✐❝✐♣❧✐♥❛s ❞❡ ▼ét♦❞♦s ◗✉❛♥t✐t❛t✐✈♦s✱ ❛í ✐♥❝❧✉í❞❛ ❛ ➪❧❣❡❜r❛ ▲✐♥❡❛r✱ ♣❡r♠✐t✐✉ ❛ r❡❞❛çã♦ ❛ q✉❛tr♦ ♠ã♦s ❞❡ss❛s ♥♦t❛s✳

❖ t❡①t♦ ♣r♦❝✉r❛ ♣r❡♣❛r❛r ♦ ❧❡✐t♦r✱ ❢✉t✉r♦ ❧✐❝❡♥❝✐❛❞♦✱ ♣❛r❛ ❜❡♠ r❡❛❧✐③❛r s❡✉ ♠✐st❡r✱ ✐❧✉str❛♥❞♦ ❛ ♣❛rt✐✲ ❝✐♣❛çã♦ ❝r❡s❝❡♥t❡ ❞❛ ♠❛t❡♠át✐❝❛ ❡♠ t♦❞❛s ❛s ❛t✐✈✐❞❛❞❡s ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❝♦❣♥✐t✐✈♦✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞❛ ✈✐sã♦ t❡ór✐❝❛ ♦✉ ❛♣❧✐❝❛❞❛✱ ❞♦ tr❛t❛♠❡♥t♦ ❡①❛t♦ ❞♦s ❝♦♥❝❡✐t♦s ❜❡♠ ❛❝❛❜❛❞♦s ♦✉ ❞♦s ♠ét♦❞♦s ❤❡✉ríst✐❝♦s ❞❡ r❡s✉❧t❛❞♦s r❡♥♦✈❛❞♦s✳

◆❡ss❡ s❡♥t✐❞♦✱ ♣r♦❝✉r❛♠♦s ❞❡s❝❛rt❛r ❛ ❛♣r❡s❡♥t❛çã♦ ✈❡rt✐❝❛❧✱ ❞❡ s❛❜♦r ❞♦❣♠át✐❝♦ ❡ ♣♦✉❝♦ ❡✜❝✐❡♥t❡✱ ❛ ♥♦ss♦ ✈❡r✱ ♥♦ ❛t✉❛❧ ❡stá❣✐♦ ❞♦ ❝✉rs♦✳ ❖♣t❛♠♦s ♣♦r ✉♠ ❡♥❢♦q✉❡ ♠❛✐s ✐♥❢♦r♠❛t✐✈♦✱ s❡♠ ❞❡s❝✉✐❞❛r✱ é ❝❧❛r♦✱ ❞❛s ❝♦♥❞✐çõ❡s ❞❡ ✈❛❧✐❞❛❞❡ ❞❛ t❡♦r✐❛ s✉❜❥❛❝❡♥t❡✳ ❚❛♠❜é♠ ♣r♦❝✉r❛♠♦s ❡①♣❧✐❝✐t❛r ♦s ♠ét♦❞♦s ♥✉♠ér✐❝♦s ❡❢❡t✐✈♦s✱ ❡✈❡♥t✉❛❧♠❡♥t❡ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❡ ❛❧❣♦r✐t♠♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s✱ ❢❡rr❛♠❡♥t❛ ✐♥❞✐s♣❡♥sá✈❡❧ ♥❛s ❛t✉❛✐s ❛♣❧✐❝❛çõ❡s✳ ❖s ❡①❡r❝í❝✐♦s ♣r♦♣♦st♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦ ♦❜❥❡t✐✈❛♠ tr❡✐♥❛r ♦s ❝♦♥❝❡✐t♦s ❛♣r❡s❡♥t❛❞♦s ❞❡ ♠♦❞♦ ❞✐r❡t♦ ❡ ❛♠✐❣á✈❡❧✳ ◆♦ss❛ ❛✜r♠❛çã♦ é ❛❜s♦❧✉t❛♠❡♥t❡ s✐♥❝❡r❛✿ ♥ã♦ ❤á ❛r♠❛❞✐❧❤❛s ✐♥t❡♥❝✐♦♥❛✐s ♥❡♠ r❡s✉❧t❛❞♦s ❞❡ ❛❧❣✐❜❡✐r❛✱ q✉❡ ♣♦❞❡r✐❛♠ ❡①✐❣✐r ❝♦♥❤❡❝✐♠❡♥t♦s ♦✉tr♦s q✉❡ ♥ã♦ ♦s ❛♣r❡s❡♥t❛❞♦s ♥❛s ♥♦t❛s✳ P♦r ❡①♣❡r✐ê♥❝✐❛ ♣ró♣r✐❛✱ ♦s ❛✉t♦r❡s ♥ã♦ ❛❝r❡❞✐t❛♠ ❡♠ s❡♠❡❧❤❛♥t❡ ♠❡t♦❞♦❧♦❣✐❛✳

●♦st❛rí❛♠♦s ❞❡ r❡❝❡❜❡r ❞❡ ✈♦❝ês✱ ♣r❡③❛❞♦s ❛❧✉♥♦s✱ s❡✉ r❡t♦r♥♦ s♦❜r❡ ❛ ❡✜❝á❝✐❛ ❡ ❛❞❡q✉❛çã♦ ❞❛s ♥♦t❛s ♣❛r❛ r❡❛❧✐③❛r ♦s ♦❜❥❡t✐✈♦s ♣r♦♣♦st♦s❀ ♦s ❡①❡r❝í❝✐♦s ❡ s✉❛s s♦❧✉çõ❡s❀ ♦ ❞❡t❛❧❤❛♠❡♥t♦ ❞❛s s♦❧✉çõ❡s✳ ❙ã♦ ✐♥❢♦r✲ ♠❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❛ ❝♦rr❡çã♦ ❞❡ r✉♠♦s ❡ ❞❡ ❝♦♥t❡ú❞♦s✳

❇♦♠ tr❛❜❛❧❤♦✦

❆♥❛ P❛✉❧❛ ▲✐♠❛ ▼❛rq✉❡s ❋❡r♥❛♥❞❡s ❆♥t♦♥✐♦ ❈❛r❧♦s ▼❛rq✉❡s ❞❛ ❙✐❧✈❛

❙✐st❡♠❛s ❧✐♥❡❛r❡s ❡ ♠❛tr✐③❡s ✺

❈❆P❮❚❯▲❖ ✶

❙■❙❚❊▼❆❙ ❉❊ ❊◗❯❆➬Õ❊❙ ▲■◆❊❆❘❊❙ ❊

▼❆❚❘■❩❊❙

❖❜❥❡t✐✈♦s ❞♦ ❈❛♣ít✉❧♦ ✶ ✭❛✮ ❘❡s♦❧✈❡r s✐st❡♠❛s ❧✐♥❡❛r❡s ♣❡❧♦ ♠ét♦❞♦ ❞❡ ●❛✉ss❀ ✭❜✮ ❈❧❛ss✐✜❝❛r ♦s s✐st❡♠❛s ❧✐♥❡❛r❡s❀

✭❝✮ ❘❡♣r❡s❡♥t❛r s✐st❡♠❛s ❧✐♥❡❛r❡s ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❡ ♠❛tr✐③❡s❀ ✭❞✮ ❊st✉❞❛r ❛s ♦♣❡r❛çõ❡s ❛❧❣é❜r✐❝❛s ❝♦♠ ♠❛tr✐③❡s❀

✭❡✮ ❈❛r❛❝t❡r✐③❛r ❛s ♠❛tr✐③❡s ✐♥✈❡rsí✈❡✐s ♣❡❧♦ ♠ét♦❞♦ ❞❡ ●❛✉ss✲❏♦r❞❛♥❀ ✭❢✮ ❉❡s❝r❡✈❡r ♦s s✐st❡♠❛s ❞❡ ❈r❛♠❡r✳

■◆❚❘❖❉❯➬➹❖

❆ ♦❝♦rrê♥❝✐❛ ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❡♠ ✈❛r✐❛❞❛s s✐t✉❛çõ❡s✱ ❞❡s❞❡ ❛♣❧✐❝❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ ✉♠ ♥ú♠❡r♦ ♣❡q✉❡♥♦ ❞❡ ❡q✉❛çõ❡s ❡ ❞❡ ✈❛r✐á✈❡✐s✱ ❛♦ tr❛t❛♠❡♥t♦ ❞❡ s✐st❡♠❛s ❞❡ ❣r❛♥❞❡ ♣♦rt❡✱ ✈❡♠ ♣r♦✈♦✲ ❝❛♥❞♦ ♦ ❛♣r✐♠♦r❛♠❡♥t♦ ❞❡ ❛❧❣♦r✐t♠♦s ♥✉♠ér✐❝♦s ❡ q✉❡ ♣♦ss❛♠ s❡r ✐♠♣❧❡♠❡♥t❛❞♦s ❡♠ ❝♦♠♣✉t❛❞♦r❡s✳ ❖s ❢✉♥❞❛♠❡♥t♦s ❞❡ss❡s ♠ét♦❞♦s t❡♠ ♦r✐❣❡♠ ♥♦ ▼➱❚❖❉❖ ❉❊ ●❆❯❙❙ ✭❞❡ ❡❧✐♠✐♥❛çã♦ s✐st❡♠át✐❝❛ ❞❡ ✈❛r✐á✈❡✐s✮✱ ♣♦st❡r✐♦r♠❡♥t❡ ❝♦♠♣❧❡♠❡♥t❛❞♦ ♣♦r ❏❖❘❉❆◆✱ ♥♦ q✉❡ ❤♦❥❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ♠ét♦❞♦ ❞❡ ●❛✉ss✲❏♦r❞❛♥✳ ❉❡ ✉♠❛ s✐♠♣❧✐❝✐❞❛❞❡ ❡st♦♥t❡❛♥t❡ ✭✐♥❝❧✉s✐✈❡ ❝♦♠♣✉t❛❝✐♦♥❛❧✦✮✱ ❡ss❡ ❛❧❣♦r✐t♠♦ ♣♦ss✉✐ ♠✉✐t♦s ♦✉tr♦s ❞❡s❞♦❜r❛♠❡♥t♦s✱ t❛♥t♦ té♦r✐❝♦s ❝♦♠♦ ❛♣❧✐❝❛❞♦s✳

✶✳✶ P❘■▼❊■❘❖❙ ❘❊❙❯▲❚❆❉❖❙

✭❛✮ ❈♦♠❡ç❛♥❞♦ ❞♦ ❝♦♠❡ç♦✱ ❥á s❛❜❡♠♦s ❝♦♠♦ ❝❛❧❝✉❧❛r ❛s ♣♦ssí✈❡✐s s♦❧✉çõ❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞♦ ♣r✐♠❡✐r♦ ❣r❛✉✱ ✐st♦ é✱ ❞❛❞♦sa❡cr❡❛✐s✱ ❡♥❝♦♥tr❛r t♦❞♦s ♦s ✈❛❧♦r❡s r❡❛✐s ❞❡x♣❛r❛ ♦s q✉❛✐s ax=c✳ ❉❡ ❢❛t♦✱ s❡ a6= 0✱ ❡♥tã♦ x =a−1

c = c

a é ❛ ú♥✐❝❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❝♦♥s✐❞❡r❛❞❛✳ P♦r ❡①❡♠♣❧♦✱ s❡ 3x= 21✱ ❡♥tã♦x = 7 é ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❛❞❛✳ ❆❣♦r❛ s❡ a= 0 ❡ c= 0✱ ❡♥tã♦ q✉❛❧q✉❡r x∈R é s♦❧✉çã♦✱ ♣♦✐s 0·x= 0 ♣❛r❛ t♦❞♦ x✳ ❊♥✜♠✱ s❡ a= 0 ❡c 6= 0✱ ❡♥tã♦ ❛ ❡q✉❛çã♦ ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦ r❡❛❧✳

✻ ■♥tr♦❞✉çã♦ à ➪❧❣❡❜r❛ ▲✐♥❡❛r ✕ ❬❆♥t♦♥✐♦ ❈❛r❧♦s ✫ ❆♥❛ P❛✉❧❛ ▼❛rq✉❡s❪

◆❡ss❡ ❝❛s♦✱ s❛❜❡♠♦s q✉❡ ❛ ❡q✉❛çã♦ ❞❛❞❛ r❡♣r❡s❡♥t❛ ✉♠❛ r❡t❛ ❞♦ ♣❧❛♥♦R2✳

P♦rt❛♥t♦✱ ❝❛❞❛ s♦❧✉çã♦(x, y) é ✉♠ ♣♦♥t♦ ❞❡ss❛ r❡t❛✱ ♦✉ s❡❥❛✱ ❤á ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s✳ ➱ ❢á❝✐❧ ❞❡s❝r❡✈❡r t♦❞❛s✳ ❙❡✱ ❞✐❣❛♠♦sa6= 0✱ ❡♥tã♦✱ ❡①♣❧✐❝✐t❛♥❞♦ ❛ ✈❛r✐á✈❡❧ x✱ t❡♠♦s x=a−1

(c−by)✱ ♦ q✉❡ ♥♦s ♣❡r♠✐t❡ ❡s❝r❡✈❡r(x, y) = (a−1

(c−by), y)✱ ♦♥❞❡ yé ❛r❜✐trár✐♦ ❡♠ R✳ ❊♥✜♠✱ t♦❞♦ ♣❛r(x, y) ❞❛ ❢♦r♠❛ ❝♦♥s✐❞❡r❛❞❛ é ♣♦♥t♦ ❞❛ r❡t❛ ax+by=c✱ ❡ r❡❝✐♣r♦❝❛♠❡♥t❡✳

P♦r ❡①❡♠♣❧♦✱ ♣❛rt✐♥❞♦ ❞❡ x+ 2y = 4✱ ✈❡♠ x = 4−2y✱ ❞♦♥❞❡ (x, y) = (4−2y, y)✱ ❡①♣r❡ssã♦ q✉❡ t❛♠❜é♠ ♣♦❞❡ s❡r ❡s❝r✐t❛(x, y) = (4,0) + (−2y, y) = (4,0) +y(−2,1)✱y∈R✱ ♦ q✉❡ ♥♦s ❞á t♦❞❛s ❛s s♦❧✉çõ❡s✳

✭❝✮ ❈♦♠♦ ♣r♦❝❡❞❡r ♥♦ ❝❛s♦ ❡♠ q✉❡ sã♦ ❞❛❞❛s ❞✉❛s ❡q✉❛çõ❡s ♥❛s ✈❛r✐á✈❡✐s r❡❛✐s (x, y)❄ ❚❛❧ ❝♦♠♦ ❛♥t❡s✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ ❝❛❞❛ ❡q✉❛çã♦ r❡♣r❡s❡♥t❛ ✉♠❛ r❡t❛ ♥♦ ♣❧❛♥♦ R2✳ ❚❡♠♦s✱ ❡♥tã♦✱ ✉♠ s✐st❡♠❛ ❞❛ ❢♦r♠❛

(

a1x+b1y=c1

a2x+b2y=c2

✱ ❡♠ q✉❡ ♦s ❝❛s♦s ❞❛s r❡t❛s s❡r❡♠ ❝♦✐♥❝✐❞❡♥t❡s ♦✉ ♣❛r❛❧❡❧❛s ♣♦❞❡♠

s❡r tr❛t❛❞♦s ❞✐r❡t❛♠❡♥t❡✳ ❉❡s❝♦♥s✐❞❡r❛♥❞♦ ❡ss❡s ❝❛s♦s✱ s❡❥❛✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ s✐st❡♠❛ (

x− y= 2 3x+ 2y= 11✳ ❖ ♠ét♦❞♦ ❣❛✉ss✐❛♥♦ ♣❡r♠✐t❡ ❡❧✐♠✐♥❛r ❛ ✈❛r✐á✈❡❧ x❞❛ s❡❣✉♥❞❛ ❡q✉❛çã♦✿ ❜❛st❛ ♠✉❧t✐♣❧✐❝❛r ❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ♣♦r −3 ❡ s♦♠❛r ♦ r❡s✉❧t❛❞♦ (−3x+ 3y = −6) ❝♦♠ ❛ s❡❣✉♥❞❛ ❡q✉❛çã♦✱ ♦❜t❡♥❞♦ ✉♠ ♦✉tr♦ s✐st❡♠❛ ✭❞❡ ❛s♣❡❝t♦ ❡s❝❛❧♦♥❛❞♦✮✱ ♠❛✐s s✐♠♣❧❡s q✉❡ ♦ s✐st❡♠❛ ♦r✐❣✐♥❛❧✿

(

x− y= 2

5y= 5✳ ❖r❛✱ ✈❡♠♦s ❡♥tã♦✱ ❞❛ ✭♥♦✈❛✮ s❡❣✉♥❞❛ ❡q✉❛çã♦ q✉❡ y = 1✳ ❊ss❡ ✈❛❧♦r ❞❡ y ❧❡✈❛❞♦ ♥❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ♥♦s ❞á x−1 = 2✱ ❞♦♥❞❡x= 3✳ ❆ss✐♠✱(x, y) = (3,1)é s♦❧✉çã♦ t❛♥t♦ ❞♦ s✐st❡♠❛ ❡s❝❛❧♦♥❛❞♦ ❝♦♠♦ ❞♦ ♦r✐❣✐♥❛❧✱ ❝♦♠♦ ✈❡♠♦s s❡♠ ❞✐✜❝✉❧❞❛❞❡✳

▼❛s ✭✳✳✳✮ s❡rá q✉❡ ♥ã♦ ❤á ♦✉tr❛ s♦❧✉çã♦❄ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❝♦♠♦ ♣♦❞❡♠♦s ❣❛r❛♥t✐r q✉❡ ♦s s✐st❡♠❛s ❞❛❞♦ ❡ ♦ ♥♦✈♦✱ ❡s❝❛❧♦♥❛❞♦✱ t❡♠ ❛s ♠❡s♠❛s s♦❧✉çõ❡s❄

P♦✐s ❜❡♠✱ ❛ ♦♣❡r❛çã♦ q✉❡ ✜③❡♠♦s ✭s♦♠❛r ❛ ✉♠❛ ❡q✉❛çã♦ ✉♠ ♠ú❧t✐♣❧♦ ❝♦♥✈❡♥✐❡♥t❡ ❞❡ ♦✉tr❛✮ ❡ ♦✉tr❛s ❞✉❛s ♦♣❡r❛çõ❡s q✉❡ ❛✐♥❞❛ ✐r❡♠♦s ❞❡s❝r❡✈❡r ✭♣❡r♠✉t❛r ❞✉❛s ❡q✉❛çõ❡s❀ ♠✉❧t✐♣❧✐❝❛r ✉♠❛ ❡q✉❛çã♦ ♣♦r ✉♠ ♥ú♠❡r♦ r❡❛❧ ♥ã♦ ♥✉❧♦✮ ♣r❡s❡r✈❛♠ ❛s s♦❧✉çõ❡s ❞♦ s✐st❡♠❛ ♦r✐❣✐♥❛❧✳ ▼❛✐s ❛✐♥❞❛✿ ❡ss❛s ♦♣❡r❛çõ❡s sã♦ t♦❞❛s r❡✈❡rsí✈❡✐s✦ ❆ q✉❛❧q✉❡r ❡stá❣✐♦ ❞❛ ❡❧✐♠✐♥❛çã♦ ❣❛✉ss✐❛♥❛✱ ♣♦❞❡♠♦s r❡t♦r♥❛r ❛♦ s✐st❡♠❛ ♦r✐❣✐♥❛❧✳ ❊ss❡s ❞♦✐s ❢❛t♦s ♥♦s ❣❛r❛♥t❡♠ q✉❡ ❛s s♦❧✉çõ❡s ❞♦s ❞♦✐s s✐st❡♠❛s sã♦ ❛s ♠❡s♠❛s✿ ♦s s✐st❡♠❛s ♦❜t✐❞♦s sã♦ s❡♠♣r❡ ❡q✉✐✈❛❧❡♥t❡s✳

❊♠ ❞❡✜♥✐t✐✈♦✱ ♦ s✐st❡♠❛ ❞❛❞♦ t❛♠❜é♠ ♣♦ss✉✐ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ (x, y) = (3,1)✳ ●❡♦♠❡tr✐❝❛✲ ♠❡♥t❡✱ ❡ss❡ ♣♦♥t♦ ❝♦rr❡s♣♦♥❞❡ à ✐♥t❡rs❡çã♦ ❞❛s r❡t❛s x−y= 2 ❡ 3x+ 2y= 11✳

❆t✐✈✐❞❛❞❡✲♣r♦♣♦st❛ ✶✳✷

✭❛✮ ❈♦♠♣❧❡t❡ ❛ ❞✐s❝✉ssã♦ ❞♦s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ❡s❜♦ç❛❞♦s ❡♠ ♥♦ q✉❛❞r♦ ✭❝✮ ❛❝✐♠❛✳ ✭❜✮ ❈♦♥s✐❞❡r❡ ♦s s✐st❡♠❛s ❞❡ ♥ú♠❡r♦s r❡❛✐s✿

(

x − y = 2 3x − 3y = 6 ✭✶✮

(

x − y = 2 x − y = 5 ✭✷✮

  

 

x − y = 2 3x + 2y = 11 4x + y = 13

✭✸✮

❱❡r✐✜q✉❡ ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ❝❛❞❛ s✐st❡♠❛✿

✭✶✮ ♣♦ss✉✐ ✐♥✜♥✐t❛s s♦❧✉çõ❡s❀ ❞❡s❝r❡✈❛ ♦ ❝♦♥❥✉♥t♦✲s♦❧✉çã♦❀ ✭✷✮ ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦❀

✭✸✮ ♣♦ss✉✐ s♦❧✉çã♦ ú♥✐❝❛✳

❙✐st❡♠❛s ❧✐♥❡❛r❡s ❡ ♠❛tr✐③❡s ✼

✶✳✸ ❖ ▼➱❚❖❉❖ ❉❊ ●❆❯❙❙

❆ ❡❧✐♠✐♥❛çã♦ ❣❛✉ss✐❛♥❛ ❝♦♥s✐st❡ ♥♦ ❛❧❣♦r✐t♠♦ q✉❡ tr❛♥s❢♦r♠❛ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ❡♠ ✉♠ ♦✉tr♦✱ ❡q✉✐✲ ✈❛❧❡♥t❡ ❛♦ ♣r✐♠❡✐r♦✱ ♠❛s ❝♦♠ ✉♠ ❢♦r♠❛t♦ ♠✉✐t♦ s✐♠♣❧❡s✱ ❞❡♥♦♠✐♥❛❞♦ s✐st❡♠❛ ❡s❝❛❧♦♥❛❞♦ ♦✉ tr✐❛♥✲ ❣✉❧❛r s✉♣❡r✐♦r✱ ❛ ♣❛rt✐r ❞♦ q✉❛❧ ❛s s♦❧✉çõ❡s ♣♦❞❡♠ s❡r ❡s❝r✐t❛s ♣♦r r❡tr♦✲s✉❜st✐t✉✐çã♦✳

❊ss❡ ❛❧❣♦r✐t♠♦ ✉s❛✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ ❛s ❝❤❛♠❛❞❛s ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s s♦❜r❡ ❛s ❡q✉❛çõ❡s ❞♦ s✐st❡♠❛S✱ ♦r✐❣✐♥❛❧♠❡♥t❡ ❞❛❞♦✱ ❛ s❛❜❡r✿

✭✶✮ ♠✉❧t✐♣❧✐❝❛r ✉♠❛ ❡q✉❛çã♦ ❞❡ S ♣♦r ✉♠ ♥ú♠❡r♦ r❡❛❧α6= 0❀

✭✷✮ s♦♠❛r ❛ ✉♠❛ ❞❛s ❡q✉❛çõ❡s ✉♠❛ ♦✉tr❛ ❡q✉❛çã♦ ♠✉❧t✐♣❧✐❝❛❞❛ ♣♦r ✉♠ ♥ú♠❡r♦ r❡❛❧❀ ✭✸✮ ♣❡r♠✉t❛r ❞✉❛s ❡q✉❛çõ❡s✳

❊①❡♠♣❧♦ ✶✳✹

P❛rt✐♥❞♦ ❞♦ s✐st❡♠❛ ❛❜❛✐①♦✱ ❡ ✐♥❞✐❝❛♥❞♦ s✉❛s ❡q✉❛çõ❡s ♣♦r E1✱ E2 ❡ E3✱❝♦♥s✐❞❡r❡♠♦s ❛s ♦♣❡r❛çõ❡s

✐♥❞✐❝❛❞❛s✿ 

 

 

x + y = 0

2x − y + 3z = 3 x −2y − z = 3

−2E1+E2 −−−−−−→

−_E1+E3   

 

x + y = 0

− 3y + 3z = 3 − 3y − z = 3

−E2+E3 −−−−−→

  

 

x + y = 0

− 3y + 3z = 3 −4z = 0

❊♥❝❡rr❛❞♦ ♦ tr❛❜❛❧❤♦ ❜r✉t♦ ❝♦♠ ❛ ♦❜t❡♥çã♦ ❞❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛✱ s❡❣✉❡ ♦ ✈❛❧♦r ❞❛s ✈❛r✐á✈❡✐s✿ ❛ ú❧t✐♠❛ ❡q✉❛çã♦ ♥♦s ❞á z = 0✱ ✈❛❧♦r q✉❡✱ ♥❛ s❡❣✉♥❞❛ ❡q✉❛çã♦✱ ❢♦r♥❡❝❡ y =−1❀ ❡♥✜♠✱ ♥❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦✱ ✈❡♠♦s q✉❡x= 1✳ ❆ss✐♠✱X= (x, y, z) = (1,−1,0)é ❛ ú♥✐❝❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❡s❝❛❧♦♥❛❞♦✳ ❚❛♠❜é♠ é ❛ ú♥✐❝❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ✐♥✐❝✐❛❧✱ ❝♦♠♦ s❡❣✉❡ ❞♦ ♣ró①✐♠♦ r❡s✉❧t❛❞♦✳

❯♠ ❞♦s ♣♦♥t♦s✲❝❤❛✈❡ ❞♦ ♠ét♦❞♦ ❞❡ ●❛✉ss é ❛ ♣r❡s❡r✈❛çã♦ ❞❛s s♦❧✉çõ❡s ❞♦s s✐st❡♠❛s ♦r✐❣✐♥❛❧S ❡ ❞♦ s✐st❡♠❛S⋆_{♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❡S♣♦r ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s s♦❜r❡ ❛s ❡q✉❛çõ❡s} ❞❡S✳ ◆❛ r❡❛❧✐❞❛❞❡✱ ♦s s✐st❡♠❛sS⋆ ❡ S ♣♦ss✉❡♠ ❛s ♠❡s♠❛s s♦❧✉çõ❡s✱ ✐st♦ é✱ sã♦ ❡q✉✐✈❛❧❡♥t❡s✳ ❆ ✈❡r✐✜❝❛çã♦ ❞❡ss❛ ♣r♦♣r✐❡❞❛❞❡ ❝♦♥s✐st❡ ❡♠ ❞♦✐s ❢❛t♦s✿

✭✶✮ ❙❡S⋆ é ♦❜t✐❞♦ ❞❡ S ♣♦r ✉♠❛ ú♥✐❝❛ ♦♣❡r❛çã♦ ❡❧❡♠❡♥t❛r✱ ❡♥tã♦✱ t♦❞❛ s♦❧✉çã♦ ❞❡ S t❛♠❜é♠ é s♦❧✉çã♦ ❞❡S⋆✳

✭✷✮ P❛r❛ ♦❜t❡r ❛ r❡❝í♣r♦❝❛✱ ✐st♦ é✱ q✉❡ t♦❞❛ s♦❧✉çã♦ ❞❡S⋆ _{t❛♠❜é♠ é s♦❧✉çã♦ ❞❡S✱ ♦❜s❡r✈❡♠♦s} q✉❡ ❝❛❞❛ ♦♣❡r❛çã♦ ❡❧❡♠❡♥t❛reé ✐♥✈❡rsí✈❡❧✱ ✐st♦ é✱ s❡etr❛♥s❢♦r♠❛ S ❡♠ S⋆✱ ♦ q✉❡ ✐♥❞✐❝❛r❡♠♦s ♣♦r S⋆=e(S)✱ ❡♥tã♦✱ s❡e1 é ❛ ♦♣❡r❛çã♦ q✉❡ r❡✈❡rt❡e✱ t❡♠♦s e1(S⋆) =e1(e(S)) =S✱ ♦✉ s❡❥❛✱ r❡♦❜t❡♠♦s

♦ s✐st❡♠❛ ✐♥✐❝✐❛❧S✳

✭✸✮ ❊♥✜♠✱ ♣♦r ❡①❡♠♣❧♦✱ s❡ ❛ ♦♣❡r❛çã♦ ❡❧❡♠❡♥t❛r e ♣❡r♠✉t❛ ❛s ❡q✉❛çõ❡s E1 ❡ E3✱ ❡♥tã♦ e1

♣❡r♠✉t❛E1❡E3❀ s❡e❝♦♥s✐st❡ ♥❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛ ❡q✉❛çã♦E2♣❡❧♦ ♥ú♠❡r♦α6= 0✱ ❡♥tã♦e1♠✉❧t✐♣❧✐❝❛

❛ ❡q✉❛çã♦E2 ♣♦r1/α❀ s❡eé ❞❛ ❢♦r♠❛ E3+αE1✱ ❡♥tã♦ e1 ❝❛❧❝✉❧❛E3−αE1✳

❊♠ r❡s✉♠♦✱ e ❡ e1 sã♦ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s ❞♦ ♠❡s♠♦ t✐♣♦✱ ✈❡r✐✜❝❛♥❞♦ e1(e(S)) =

e(e1(S))✳

✽ ■♥tr♦❞✉çã♦ à ➪❧❣❡❜r❛ ▲✐♥❡❛r ✕ ❬❆♥t♦♥✐♦ ❈❛r❧♦s ✫ ❆♥❛ P❛✉❧❛ ▼❛rq✉❡s❪

❊①❡♠♣❧♦ ✶✳✺

❉❡♣♦✐s ❞❡ ❡s❝❛❧♦♥❛r ❝❛❞❛ s✐st❡♠❛✱ ✐♥❞✐q✉❡✱ s❡ ♣♦ssí✈❡❧✱ t♦❞❛s ❛s s♦❧✉çõ❡s❀ ❝❧❛ss✐✜q✉❡ ♦s s✐st❡♠❛s✳ ✭✶✮

(

x+ y= 2

5y= 10✳ ❙✐st❡♠❛ ❥á ❡s❝❛❧♦♥❛❞♦✳ ❉❛ s❡❣✉♥❞❛ ❡q✉❛çã♦✱ ✈❡♠♦s q✉❡y= 2✱ ❞♦♥❞❡✱ ❞❛ ♣r✐♠❡✐r❛✱ x= 0✳ ❖ s✐st❡♠❛ ♣♦ss✉✐ s♦❧✉çã♦ ú♥✐❝❛ X= (0,2)❡ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛❞♦✳

✭✷✮     

x+ y + z= 1

x+ 2y + 2z= 1 2x+ 3y + 3z= 2

−E1+E2 −−−−−−→

−2E1+E3   

 

x+y +z= 1

y +z= 0

y +z= 0

−E2+E3 −−−−−−→     

x+y+z= 1

y+z= 0 0 = 0

− →

(

x+y+z= 1

y+z= 0

❆s ✈❛r✐á✈❡✐s ❧✐❞❡r❡s sã♦ x ❡ y❀z é ✉♠❛ ✈❛r✐á✈❡❧ ❧✐✈r❡✳ ❈❛❞❛ s♦❧✉çã♦ s❡rá ❡①♣❧✐❝✐t❛❞❛ ❡♠ ❢✉♥çã♦ ❞❡z✳ ❈♦♠♦y+z= 0✱ ✈❡♠♦s q✉❡x= 1 ❡y=−z✱ ❞♦♥❞❡X = (x, y, z) = (1,−z, z) = (1,0,0) +z(0,−1,1)✱ z∈R✳ ❖ s✐st❡♠❛ ❞❛❞♦ é ♣♦ssí✈❡❧ ✐♥❞❡t❡r♠✐♥❛❞♦✿ ❤á ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s✳ ❖❜s❡r✈❡♠♦s q✉❡ X ❞❡s❝r❡✈❡ ✉♠❛ r❡t❛ ❞♦ R3✱ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦(1,0,0)❡ é ♣❛r❛❧❡❧❛ ❛♦ ✈❡t♦r(0,−1,1)✳

✭✸✮     

x+ y+ z= 1 x+ 2y+ 2z= 1 2x+ 3y+ 3z= 4

−E1+E2 −−−−−−→

−2E1+E3   

 

x+y+z= 1 y+z= 0 y+z= 2

−E2+E3 −−−−−→     

x+y+z= 1 y+z= 0 0 = 2

❖ s✐st❡♠❛ é ✐♠♣♦ssí✈❡❧❀ ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦ r❡❛❧✳ ❖❜s❡r✈❡ ♦ s✐st❡♠❛ ❝❡♥tr❛❧✿ ❛s ❡q✉❛çõ❡sE2 ❡E3sã♦✱

♠❛♥✐❢❡st❛♠❡♥t❡✱ ✐♥❝♦♠♣❛tí✈❡✐s✳ ▼❡s♠♦ q✉❡ ✐ss♦ t❡♥❤❛ ♣❛ss❛❞♦ ❞❡s❛♣❡r❝❡❜✐❞♦✱ ♦ s✐st❡♠❛ ❞❛ ❞✐r❡✐t❛ ♥ã♦ ❞❡✐①❛ ❞ú✈✐❞❛s✱ q✉❛♥❞♦ ❛♥✉♥❝✐❛ q✉❡0 = 2 ✭✦✮✳

❆t✐✈✐❞❛❞❡✲♣r♦♣♦st❛ ✶✳✻

✭✶✮ ❯s❡ ♦ ❡s❝❛❧♦♥❛♠❡♥t♦ ❣❛✉ss✐❛♥♦ ♣❛r❛ ❛❝❤❛r t♦❞❛s ❛s s♦❧✉çõ❡s ❡ ❝❧❛ss✐✜❝❛r ♦s s✐st❡♠❛s✳ (

2x + 3y = 13 x − y = −1 ✭✐✮

  

 

x + 3y = 1 2x + y =−3 x + y = 0

✭✐✐✮     

x + 2y = 4 y − z = 0 x + 2z = 4

✭✐✐✐✮     

4x + y − z = 1 2x + 2y + z = 5 x − y − z = −4

✭✐✈✮

✭✷✮ ❈♦♥s✐❞❡r❡ ♦ s✐st❡♠❛ (

x − y = 1

3x −3y = α ✱ ♦♥❞❡α é ✉♠ ♣❛râ♠❡tr♦ r❡❛❧✳

▼♦str❡ q✉❡✱ s❡ α6= 3✱ ❡♥tã♦ ♦ s✐st❡♠❛ é ✐♠♣♦ssí✈❡❧❀ s❡α= 3✱ ♦ s✐st❡♠❛ é ♣♦ssí✈❡❧ ✐♥❞❡t❡r♠✐♥❛❞♦✳ ✭✸✮ ❊♥❝♦♥tr❡ ❛ ♣❛rá❜♦❧❛f(x) =ax2+bx+cq✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s (1,2)✱(−1,6)❡(2,3)✳

✶✳✼ ❘❊❙❯▼❖✿ ❘❡s♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r

P❛rt✐♥❞♦ ❞❡ ✉♠ s✐st❡♠❛S✱ s❡❥❛S⋆ _{♦ ❝♦rr❡s♣♦♥❞❡♥t❡ s✐st❡♠❛ ❡s❝❛❧♦♥❛❞♦✳ P♦❞❡♠ ♦❝♦rr❡r três ❝❛s♦s✿}

✭✶✮S⋆ _{é ❞❛ ❢♦r♠❛}           

a1x+b1y +c1z=d1

b2y +c2z=d2

c3z=d3

0 = 0 →❡q✉❛çõ❡s r❡❞✉♥❞❛♥t❡s

❉❡s❝❛rt❛❞❛s ❛s ❡q✉❛çõ❡s r❡❞✉♥❞❛♥t❡s 0 = 0✱ r❡st❛ ✉♠ s✐st❡♠❛ ✏q✉❛❞r❛❞♦✑✱ ❝♦♠ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❡q✉❛çõ❡s ❡ ✈❛r✐á✈❡✐s ❡ ❝♦♥t❡♥❞♦ ❛♣❡♥❛s ✈❛r✐á✈❡✐s ❧í❞❡r❡s ❛♦ ❧♦♥❣♦ ❞❛ ✏❞✐❛❣♦♥❛❧✑✳ P♦r r❡tr♦ s✉❜st✐t✉✐çã♦✱ ✈❡♠♦s q✉❡ ♦ s✐st❡♠❛ ♣♦ss✉✐ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ X= (x, y, z)❀ s✐st❡♠❛ ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛❞♦✳

✭✷✮ ❉❡s❝❛rt❛❞❛s ❛s r❡❞✉♥❞â♥❝✐❛s 0 = 0✱ ♦ s✐st❡♠❛ ❡s❝❛❧♦♥❛❞♦ é ❞❛ ❢♦r♠❛ (

a1x+b1y+c1z=d1

b2y+c2z=d2

❙✐st❡♠❛s ❧✐♥❡❛r❡s ❡ ♠❛tr✐③❡s ✾

Pr❛t✐q✉❡ ✉♠ ♣♦✉❝♦✳

❈♦♠♣r♦✈❡ ❛ ❞✐s❝✉ssã♦ ❛♥t❡r✐♦r✱ ✈❡r✐✜❝❛♥❞♦✱ ❞❡t❛❧❤❛❞❛♠❡♥t❡✱ ♦s r❡s✉❧t❛❞♦s ♥✉♠ér✐❝♦s ❞♦s s✐st❡♠❛s ❥á r❡s♦❧✈✐❞♦s✳

✶✳✽ ❘❊P❘❊❙❊◆❚❆➬➹❖ ▼❆❚❘■❈■❆▲ ❉❊ ❙■❙❚❊▼❆❙ ▲■◆❊❆❘❊❙

❍á ✉♠❛ ❢♦rt❡ ❝♦♥❡①ã♦ ❡♥tr❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ❡ ♠❛tr✐③❡s✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s sã♦ ✐♠❜r✐❝❛❞❛s ❡ r❡✈❡❧❛♠ ❡①tr❛♦r❞✐♥ár✐❛ r❡♣❡r❝✉ssã♦ ♥❛s ❞✉❛s ❝❛t❡❣♦r✐❛s ❞❡ ♦❜❥❡t♦s✳

❈♦♠♦ s❛❜❡♠♦s✱ ♠❛tr✐③❡s ❞❡ ♥ú♠❡r♦s r❡❛✐s sã♦ q✉❛❞r♦s ♦r❞❡♥❛❞♦s ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ ❞✐s♣♦st♦s ❡♠ ❧✐♥❤❛s ❡ ❡♠ ❝♦❧✉♥❛s✳ ❙❡ A ❞❡♥♦t❛ ✉♠ t❛❧ q✉❛❞r♦✱ q✉❡ ♣♦ss✉✐ m ❧✐♥❤❛s ❡ n ❝♦❧✉♥❛s✱ ❞✐r❡♠♦s q✉❡ ❛ ♠❛tr✐③ A é ❞❡ ♦r❞❡♠ m×n ❡ ✐♥❞✐❝❛r❡♠♦s Am×n✳ ◆❛ ♣❛rt❡ ✐♥✐❝✐❛❧ ❞❡ss❡ tr❛❜❛❧❤♦✱ q✉❛s❡ s❡♠♣r❡ ✉s❛r❡♠♦s1≤m, n≤3✳

P♦r ❡①❡♠♣❧♦✿

A=

"

1 −1

3 2

#

é ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛2×2✱B =

   1 3 2 1 1 1  

é ✉♠❛ ♠❛tr✐③3×2✱C= 

 

1 1 0

2 −1 3 1 −2 −1





é ✉♠❛ ♠❛tr✐③

q✉❛❞r❛❞❛3×3✱ D=

"

1 1 0

2 1 −3

#

é ✉♠❛ ♠❛tr✐③2×3✱X =

   1 −1 0  

é ✉♠❛ ♠❛tr✐③3×1✳

◆♦ss♦ ♦❜❥❡t✐✈♦ ✐♠❡❞✐❛t♦ é ❛ss♦❝✐❛r ❛ ✉♠ ❞❛❞♦ s✐st❡♠❛ s✉❛ r❡♣r❡s❡♥t❛çã♦ ♠❛tr✐❝✐❛❧✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ 

 

 

a1x + b1y + c1z =d1

a2x + b2y + c2z =d2

a3x + b3y + c3z =d3

→ A= 

 

a1 b1 c1

a2 b2 c2

a3 b3 c3



  , X =

   x y z    , Y =

   d1 d2 d3  

 → AX=Y

❆ ♠❛tr✐③A é ❛ ♠❛tr✐③ ❞♦s ❝♦❡✜❝✐❡♥t❡s✱X é ♦ ✈❡t♦r ✭♠❛tr✐③3×1✮ ❞❛s ✈❛r✐á✈❡✐s ❡ Y ♦ ✈❡t♦r s❡❣✉♥❞♦✲ ♠❡♠❜r♦✳ ❖ ♣r♦❞✉t♦ AX é ♦ ♣r♦❞✉t♦✲❧✐♥❤❛✲♣♦r✲❝♦❧✉♥❛ ✭♥❛ r❡❛❧✐❞❛❞❡✱ ♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r✮✱ ♦♥❞❡ ❝❛❞❛ ❧✐♥❤❛ ❞❡ ❆✱ ❞✐❣❛♠♦s [a1 b1 c1]é ♠✉❧t✐♣❧✐❝❛❞❛ ♣❡❧♦ ✈❡t♦r X✱ ❢♦r♥❡❝❡♥❞♦ ❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ a1x+

b1y+c1z✱ ♣♦st❡r✐♦r♠❡♥t❡ ✐❣✉❛❧❛❞❛ ❛ d1✱ ✐st♦ é✱a1x+b1y+c1z=d1✳

◆ã♦ é ❞✐❢í❝✐❧ ✐♠❛❣✐♥❛r ❛ s✐♠♣❧✐✜❝❛çã♦ q✉❡ t❡♠♦s ❡♠ ♠❡♥t❡✿ ❛♦ ❡s❝❛❧♦♥❛r ✉♠ s✐st❡♠❛ ❧✐♥❡❛r✱ ♦♣❡r❛♠♦s s♦❜r❡ s✉❛s ❡q✉❛çõ❡s✱ ✐st♦ é✱ s♦❜r❡ ❛s ❧✐♥❤❛s ❞❛ ♠❛tr✐③ A ❞❡ s❡✉s ❝♦❡✜❝✐❡♥t❡s✳ ❙ó q✉❡ é ♥❡❝❡ssár✐♦ ❢❛③❡r ♦ s❡❣✉♥❞♦ ♠❡♠❜r♦Y r❡❝❡❜❡r ❛s ♠❡s♠❛s ♦♣❡r❛çõ❡s✦ ❈♦♠♦ ♣r♦❝❡❞❡r❄ ➱ só ❛❝r❡s❝❝❡♥t❛rY ❝♦♠♦ ♠❛✐s ✉♠❛ ❝♦❧✉♥❛ ❞❡A✱ ♦ q✉❡ ♥♦s ❞á ❛ ❝❤❛♠❛❞❛ ♠❛tr✐③ ❛✉♠❡♥t❛❞❛ Aˆ✳

  

 

a1x + b1y + c1z =d1

a2x + b2y + c2z =d2

a3x + b3y + c3z =d3

→ A= 

 

a1 b1 c1

a2 b2 c2

a3 b3 c3





 → Aˆ= 

 

a1 b1 c1 d1

a2 b2 c2 d2

a3 b3 c3 d3



 

❘❡t♦♠❛♥❞♦ ♦ ❊①❡♠♣❧♦ ✶✳✹✿ 

 

 

x + y = 0

2x − y + 3z = 3 x −2y − z = 3

→Aˆ= 

 

1 1 0 0 2 −1 3 3 1 −2 −1 3



 

−2L1+L2 −−−−−−→

−L1+L3 

 

1 1 0 0 0 −3 3 3 0 −3 −1 3



 

−L2+L3 −−−−−→



 

1 1 0 0 0 −3 3 3 0 0 −4 0



 ✳

❖❜t✐❞♦ ♦ ❡s❝❛❧♦♥❛♠❡♥t♦✱ s❡❣✉❡ ❛ s♦❧✉çã♦ ♣♦r r❡tr♦✲s✉❜st✐t✉✐çã♦✳

❆t✐✈✐❞❛❞❡✲♣r♦♣♦st❛ ✶✳✾

✶✵ ■♥tr♦❞✉çã♦ à ➪❧❣❡❜r❛ ▲✐♥❡❛r ✕ ❬❆♥t♦♥✐♦ ❈❛r❧♦s ✫ ❆♥❛ P❛✉❧❛ ▼❛rq✉❡s❪

✶✳✶✵ ❆♣❧✐❝❛çã♦ ❛♦s s✐st❡♠❛s ❤♦♠♦❣ê♥❡♦s

❯♠ s✐st❡♠❛ ❧✐♥❡❛r é ❤♦♠♦❣ê♥❡♦ q✉❛♥❞♦ é ❞❛ ❢♦r♠❛ AX = 0✳ ❯♠ t❛❧ s✐st❡♠❛ ♥✉♥❝❛ é ✐♠♣♦ssí✈❡❧✱ ❥á q✉❡✱ ♣❡❧♦ ♠❡♥♦s✱ ♣♦ss✉✐ ❛ ❝❤❛♠❛❞❛ s♦❧✉çã♦ tr✐✈✐❛❧ X = 0✳ ◆❛ r❡❛❧✐❞❛❞❡✱ ❤á ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ❡♠ ❞❡❝✐❞✐r q✉❛♥❞♦ ♦ s✐st❡♠❛ ♣♦ss✉✐ ❛♣❡♥❛s ❛ s♦❧✉çã♦ tr✐✈✐❛❧✱ ♦✉ q✉❛♥❞♦ ♣♦ss✉✐ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s✳ ❊♠ q✉❛❧q✉❡r ❝❛s♦✱ ❞❛❞❛ Am×n✱ ♦ ❝♦♥❥✉♥t♦ ❞❛s s♦❧✉çõ❡s {X ∈Rn ; AX =O} é ♦ ♥ú❝❧❡♦ ❞❡ A✱ ♥♦t❛❞♦N(A)✳

❊①❡♠♣❧♦ ✶✳✶✶

◆♦ s✐st❡♠❛ AX= 0 ❛❜❛✐①♦✱ ♦ ❡s❝❛❧♦♥❛♠❡♥t♦ r❡✈❡❧❛ ❛♣❡♥❛s ❛ s♦❧✉çã♦ tr✐✈✐❛❧X = 0✳

A= 

 

1 1 0 2 −1 3 1 −2 −1



 

−2L1+L2 −−−−−−→

−L1+L3 

 

1 1 0

0 −3 3 0 −3 −1



 

−L2+L3 −−−−−→



 

1 1 0

0 −3 3 0 0 −4



 

−(1/3)L2 −−−−−−→

−(1/4)L3 

 

1 1 0 0 1 −1 0 0 1



 

❆t✐✈✐❞❛❞❡✲♣r♦♣♦st❛ ✶✳✶✷

❊♥❝♦♥tr❡ t♦❞❛s ❛s s♦❧✉çõ❡s ❞❡ ❝❛❞❛ s✐st❡♠❛ ❤♦♠♦❣ê♥❡♦AX = 0✿

✭✶✮A= "

1 1 1 −1

#

❀ ✭✷✮ A= 

 

1 1 1 −1 1 2





❀ ✭✸✮A= "

1 1 0

2 −1 −3 #

✭✹✮ ▼♦str❡ q✉❡ ❛ ♦♣❡r❛çã♦ ❡❧❡♠❡♥t❛r ❞❡ ♣❡r♠✉t❛r ❞✉❛s ❧✐♥❤❛s ♣♦❞❡ s❡r r❡❛❧✐③❛❞❛ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❛s ❞✉❛s ♦✉tr❛s ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s✳

✶✳✶✸ ❖P❊❘❆➬Õ❊❙ ❈❖▼ ▼❆❚❘■❩❊❙ ✭❛✮ ❙♦♠❛

❉✉❛s ♠❛tr✐③❡s A❡ B ❞❡ ♠❡s♠❛ ♦r❞❡♠ ♣♦❞❡♠ s❡r s♦♠❛❞❛s✿ ❛ s♦♠❛A+B é ✉♠❛ ♠❛tr✐③✱ ❛✐♥❞❛ ❞❛ ♠❡s♠❛ ♦r❞❡♠ ❞❡ A ❡B✱ ♦❜t✐❞❛ ♣❡❧❛ s♦♠❛ ❞♦s ❡❧❡♠❡♥t♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❞❡A ❡ ❞❡ B✳

✭❜✮ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦r ✉♠❛ ♠❛tr✐③

❉❛❞♦s α∈R❡ ❛ ♠❛tr✐③ Am×n✱ ♥♦t❛r❡♠♦s α A❛ ♠❛tr✐③m×n ♦❜t✐❞❛ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r α ❞❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡ A✳

❱❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✭✶✮ ❆ s♦♠❛ é ❛ss♦❝✐❛t✐✈❛✿ ✭A ✰B✮ ✰ C❂A✰✭B✰C✮❀

✭✷✮ ❊①✐stê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❛❞✐t✐✈♦✳ ❉❛❞❛ Am×n✱ ❡♥tã♦ ❛ ♠❛tr✐③ ♥✉❧❛ O_m×_n ✭❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ t♦❞♦s ✐❣✉❛✐s ❛ ③❡r♦✮ é ❛ ú♥✐❝❛ ♠❛tr✐③ t❛❧ q✉❡A+O=O+A=A✳

✭✸✮ ❊①✐stê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦ ♦♣♦st♦ ❛❞✐t✐✈♦✳ ❉❛❞❛ A✱ ❛ ♠❛tr✐③−A✱ ♦❜t✐❞❛ ❞❡ A♣❡❧♦ ♣r♦❞✉t♦ ❞❡ ❝❛❞❛ ✉♠ ❞❡ s❡✉s ❡❧❡♠❡♥t♦s ♣♦r−1✱ é ❛ ú♥✐❝❛ ♠❛tr✐③ t❛❧ q✉❡ A+ (−A) = (−A) +A=O✳

✭✹✮ ❆ s♦♠❛ é ❝♦♠✉t❛t✐✈❛✿ A+B =B+A✳

✭✺✮ ❉✐str✐❜✉t✐✈✐❞❛❞❡ ❡♠ r❡❧❛çã♦ à s♦♠❛ ❞❡ ♠❛tr✐③❡s✿ α(A+B) =αA+αB ✭✻✮ ❉✐str✐❜✉t✐✈✐❞❛❞❡ ❡♠ r❡❧❛çã♦ à s♦♠❛ ❞❡ r❡❛✐s✿ (α+β)A=αA+βB ✭✼✮ ❆ss♦❝✐❛t✐✈✐❞❛❞❡✿ α(βA) = (αβ)A

✭✽✮ ■♥✈❛r✐â♥❝✐❛ ♣♦r1∈R✿ 1·A=A✳

❙✐st❡♠❛s ❧✐♥❡❛r❡s ❡ ♠❛tr✐③❡s ✶✶

❊①❡♠♣❧♦ ✶✳✶✹

❙❡❥❛♠A= "

2 1 3 −2 0 0

# ❡ B=

" 8 1 3 0 4 2

#

✳ ❚❡♠♦s✱ s✉❝❡ss✐✈❛♠❡♥t❡✿

✭✶✮A+B = "

2 1 3 −2 0 0 #

+ "

8 1 3 0 4 2

# =

"

10 2 6 −2 4 2

#

✭✷✮3A−B = "

6 3 9 −6 0 0 #

− "

8 1 3 0 4 2

# =

"

−2 2 6 −6 −4 −2

#

✭✸✮ ❙❡ ❛ ♠❛tr✐③X é t❛❧ q✉❡6X+ 3A= 4X+B✱ ❡♥tã♦ 2X=B−3A❀ ❧♦❣♦✱ t❡♠♦sX= "

1 −1 −3 3 2 1

# .

✭❝✮ Pr♦❞✉t♦ ❞❡ ♠❛tr✐③❡s

❖ ♣r♦❞✉t♦AB ❞❛s ♠❛tr✐③❡s✱ Am×n ❡ Bn×p✱ ♥❡ss❛ ♦r❞❡♠✱ é ✉♠ t✐♣♦ ❞❡ ♣r♦❞✉t♦ ❧✐♥❤❛ ❞❡A ♣♦r ❝♦❧✉♥❛ ❞❡B✱ ♦ q✉❡ ❡①✐❣❡ q✉❡ ❝❛❞❛ ❧✐♥❤❛ ❞❡At❡♥❤❛ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❛s ❝♦❧✉♥❛s ❞❡B✱ ❞♦♥❞❡ ❛s ♦r❞❡♥s ❛❝✐♠❛ ❝♦♥s✐❞❡r❛❞❛s❀m ❡ p✐♥❞✐❝❛♠ ♦ t❛♠❛♥❤♦ ❞♦ ♣r♦❞✉t♦ ABm×p✳

❖✉tr❛s ♣r♦♣r✐❡❞❛❞❡s

✭✶✮ ❉✐str✐❜✉t✐✈✐❞❛❞❡s✿ A(B+C) =AB+AC❀(A+B)C =AC+BC ✭✷✮ ❆ss♦❝✐❛t✐✈✐❞❛❞❡✿ A(BC) = (AB)C

❊①❡♠♣❧♦ ✶✳✶✺

✭✶✮ ❆s ♠❛tr✐③❡sA2×3 ❡ B3×4✱ ♥❡ss❛ ♦r❞❡♠✱ ♣♦❞❡♠ s❡r ♠✉❧t✐♣❧✐❝❛❞❛s ❡AB2×4❀ ❡♥tr❡t❛♥t♦✱ ♦ ♣r♦❞✉t♦ BA♥ã♦ ❡stá ❞❡✜♥✐❞♦✳

✭✷✮ ❆s ♠❛tr✐③❡sA2×3❡C3×2 ♣♦❞❡♠ s❡r ♠✉❧t✐♣❧✐❝❛❞❛s ♥❛ ♦r❞❡♠AC2×2 ❡ ♥❛ ♦r❞❡♠CA3×3❀ ❡♥tr❡t❛♥t♦✱ ❡✈✐❞❡♥t❡♠❡♥t❡✱AC 6=CA✳

✭✸✮ ❆♣❡♥❛s ♥♦ ❝❛s♦ ❞❡ ♠❛tr✐③❡s A, B q✉❛❞r❛❞❛s ❡ ❞❡ ♠❡s♠❛ ♦r❞❡♠✱ ♣♦❞❡r✐❛♠ ❝♦✐♥❝✐❞✐r ♦s ♣r♦❞✉t♦s AB ❡BA✳ ▼❛s✱ ❡♠ ❣❡r❛❧✱ ♠❡s♠♦ ♥❡ss❡ ❝❛s♦✱AB6=BA✳ ❖❜s❡r✈❡ ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✳

" 1 0 1 0

# ·

" 0 0 1 1

# =

" 0 0 0 0

# 6=

" 0 0 2 0

# =

" 0 0 1 1

# ·

" 1 0 1 0

#

❊♠ ❞❡✜♥✐t✐✈♦✱ ♦ ♣r♦❞✉t♦ ❞❡ ♠❛tr✐③❡s ♥ã♦ é ❝♦♠✉t❛t✐✈♦✳

✭✹✮ ❖ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r t❛♠❜é♠ ♠♦str❛ ✉♠ ♣r♦❞✉t♦AB=O✱ ❝♦♠A6=O ❡ B 6=O✳

✭✺✮ ❙❡ A é q✉❛❧q✉❡r ♠❛tr✐③✱ ❡♥tã♦ AO = O ❡ OA = O✳ ❈✉✐❞❛❞♦ ❝♦♠ ♦ ❛❜✉s♦ ❞❡ ♥♦t❛çã♦✱ q✉❡ r❡♣r❡s❡♥t❛ ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❛tr♦ ♠❛tr✐③❡s ♥✉❧❛s ❞✐st✐♥t❛s✳ P♦r ❡①❡♠♣❧♦✱ ❡♠AO=O ❛s ♦r❞❡♥s sã♦ Am×n ❡ On×p✱ ❞♦♥❞❡ ♦ s❡❣✉♥❞♦ ♠❡♠❜r♦ é ❞❡ ♦r❞❡♠ Om×p✳ ❆♣❡♥❛s ♥♦ ❝❛s♦ A ❡ O q✉❛❞r❛❞❛s ❞❡ ♠❡s♠❛ ♦r❞❡♠✱ ✈❛❧❡AO=OA=O✳

✶✳✶✻ ■♥✈❛r✐â♥❝✐❛ ♣❡❧❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡✳

❆ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ■ é ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❡♠ q✉❡ ♦s ❡❧❡♠❡♥t♦s ❞❛ ❞✐❛❣♦♥❛❧ ✭♣r✐♥❝✐♣❛❧✮ sã♦ ✐❣✉❛✐s ❛ ✶✱ ❡ sã♦ ♥✉❧♦s t♦❞♦s ♦s ❞❡♠❛✐s ❡❧❡♠❡♥t♦s✳ ❙❡ A é q✉❛❧q✉❡r ♠❛tr✐③✱ ❡♥tã♦ AI =A ❡ IA= A✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡A ❡ I sã♦ q✉❛❞r❛❞❛s ❞❡ ♠❡s♠❛ ♦r❞❡♠✱ ❡♥tã♦AI =IA=A✳

✶✷ ■♥tr♦❞✉çã♦ à ➪❧❣❡❜r❛ ▲✐♥❡❛r ✕ ❬❆♥t♦♥✐♦ ❈❛r❧♦s ✫ ❆♥❛ P❛✉❧❛ ▼❛rq✉❡s❪

❖❜s❡r✈❛çã♦ ❱ár✐❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ♦♣❡r❛çõ❡s ❝♦♠ ♥ú♠❡r♦s r❡❛✐s t❛♠❜é♠ ❝♦♠♣❛r❡❝❡♠ ♥❛s ♠❛✲ tr✐③❡s✳ ➱ ♦ ❝❛s♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❞❛s ♣♦tê♥❝✐❛s ❞❡ ❡①♣♦❡♥t❡ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦p≥1♣❛r❛ ♠❛tr✐③❡s q✉❛❞r❛❞❛s An×n✿ A1=A❀Ap =A×A× · · · ×A

| {z }

p

❊①❡♠♣❧♦ ✶✳✶✼

✭✶✮ ❊♠ ❝❛❞❛ ❝❛s♦✱ ❛❝❤❡ t♦❞❛s ❛s ♣♦tê♥❝✐❛s A2, A3, . . . , Ap, . . . ✭p≥1 ✐♥t❡✐r♦✮✿

✭❛✮A= "

1 a 0 1

#

✭❜✮A= "

1 0 0 −1

#

✭❝✮A= 

 

0 1 1 0 0 1 0 0 0



 

◆❡ss❡ ❡①❡♠♣❧♦✱ ✉♠❛ ✐❞é✐❛ é ❝❛❧❝✉❧❛r ♣♦tê♥❝✐❛s s✉❝❡ss✐✈❛s A2, A3, . . . ❛té q✉❡ s❡ ♣❡r❝❡❜❛ ✉♠❛ ❧❡✐ ❞❡ ❢♦r♠❛çã♦✳

✭❛✮ ❚❡♠♦sA2 ₌

" 1 a 0 1 # " 1 a 0 1 # = " 1 2a 0 1 #

❀ ❡♠ s❡❣✉✐❞❛✱A3 _=A2 _A=

" 1 2a 0 1 # " 1 a 0 1 # = " 1 3a 0 1 #

✱ s✉❣❡r✐♥❞♦ q✉❡ Ap= "

1 pa 0 1

#

✱ r❡s✉❧t❛❞♦ q✉❡ ♣♦❞❡ s❡r ✈❡✜❝❛❞♦ ♣♦r ✐♥❞✉çã♦ s♦❜r❡p✳

✭❜✮ ❯s❛♥❞♦ ♦ r❡s✉❧t❛❞♦ A2 _{=I ✭✈❡r✐✜q✉❡✦✮✱ ✈❡♠ A}3_=A2_A=IA=A✱A4 _=A2_·A2 _{=I·I =I✱ ❡t❝✳}

❆ss✐♠✱ s❡p >0 é ✉♠ ✐♥t❡✐r♦ ♣❛r t❡♠♦s Ap =I✱ ❡ s❡ p é í♠♣❛r✱ ❡♥tã♦Ap =A✳ ✭❝✮ ❱❡r✐✜q✉❡ q✉❡ A2 _{6=O ❡ A}3 _{=O✱ ❞♦♥❞❡ A}p _{=O s❡p≥3✳}

✭✷✮ ❙❡❥❛♠ ❛s ♠❛tr✐③❡s A, B, C✱ t❛✐s q✉❡A6=O ❡ AB=AC❀ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡B =C❄

❈♦♠♦AB=AC⇐⇒A(B−C) =O✱ ❜❛st❛ ❝♦♥s✐❞❡r❛r ♦ ❊①❡♠♣❧♦ ✶✳✶✺ ✭✹✮✿ ❡s❝♦❧❤❡♠♦s ❛s ♠❛tr✐③❡s

A= "

1 0 1 0

#

✱B−C= "

0 0 1 1

#

✱ ❡B 6=C✳

✭✸✮ ❬❚r❛♥s♣♦s✐çã♦❪ ❉❛❞❛ ✉♠❛ ♠❛tr✐③ Am×n✱ ❛ ♠❛tr✐③ tr❛♥s♣♦st❛ tA ❞❡ A é ❛ ♠❛tr✐③ n×m ♦❜t✐❞❛ ♣❡❧❛ tr♦❝❛ ♦r❞❡♥❛❞❛ ❞❛s ❧✐♥❤❛s ✭❝♦❧✉♥❛s✮ ♣❡❧❛s ❝♦❧✉♥❛s ✭❧✐♥❤❛s✮ ❞❡ A✳ ❆ss✐♠✱ ♣♦r ❡①❡♠♣❧♦✿

A=    1 2 0 3 4 1  

 =⇒ tA= "

1 0 4 2 3 1

#

❀X =

   1 0 4  

 =⇒ tX = h

1 0 4i❀ P =

" a b b a

#

=⇒ t_{P =} "

a b b a

#

✳

Pr♦♣r✐❡❞❛❞❡s❱❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ❞❛ tr❛♥s♣♦s✐çã♦ ✭✈❡r✐✜q✉❡✦✮✿

✭❛✮t₍t_{A) =A❀}

✭❜✮t_{(A+B) =}t_A+t_B❀ ✭❝✮t_{(αA) =α} t_A❀

✭❞✮t_{(AB) =}t_B t_{A ✭❛t❡♥çã♦ à ♦r❞❡♠ ❞♦s ❢❛t♦r❡s✦✮}

❆t✐✈✐❞❛❞❡✲♣r♦♣♦st❛ ✶✳✶✽ ✭✶✮ ❉❛❞❛s ❛s ♠❛tr✐③❡s

A=

"

1 3

−4 2

#

❀ B=

"

0 −2 1

−1 3 0

#

❀ C=



 

−2 0 1

0 3 0

2 3 −1



 ❀ D=



 

2 −4

0 0

3 3





❀ E= "

1 −1

1 0

#

✱

❝❛❧❝✉❧❡ ❛s s❡❣✉✐♥t❡s ♠❛tr✐③❡s

❙✐st❡♠❛s ❧✐♥❡❛r❡s ❡ ♠❛tr✐③❡s ✶✸

✭✷✮ ❆❝❤❡ t♦❞❛s ❛s ♠❛tr✐③❡sX q✉❡ ❝♦♠✉t❛♠ ❝♦♠ ❛ ♠❛tr✐③A= "

1 1 0 0

# ✳

✭✸✮ ❈♦♥s✐❞❡r❡ ♦ s✐st❡♠❛AX=Y✱A=



 

1 2 −3 3 −1 −2 1 −5 4



 ✱X=



  x y z



  ❡Y =



  a b c



 ✳

✭✐✮ ■♥❞✐q✉❡ ❛ r❡❧❛çã♦ ❡♥tr❡a✱b❡ c♣❛r❛ ❛ q✉❛❧ ♦ s✐st❡♠❛ ♣♦ss✉✐ s♦❧✉çã♦❀

✭✐✐✮ ❊♥❝♦♥tr❡ t♦❞❛s ❛s s♦❧✉çõ❡s ❞♦ s✐st❡♠❛ ❞❛❞♦ s❡Y =



 

−2 1 5



 ❀Y =



 

1 1 1



 ✳

✶✳✶✾ ▼❆❚❘■❩❊❙ ■◆❱❊❘❙❮❱❊■❙

❉❛❞❛ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A✱ ❞✐r❡♠♦s q✉❡ A é ✐♥✈❡rsí✈❡❧ s❡ ❡①✐st❡ ✉♠❛ ♠❛tr✐③ B t❛❧ q✉❡ AB = BA = I✳ ◆❛ r❡❛❧✐❞❛❞❡✱ ✉♠❛ t❛❧ ♠❛tr✐③ B é ✉♥✐❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛ ♣♦r A❀ ❞❡ ❢❛t♦✱ s❡ C é ❛❧❣✉♠❛ ♠❛tr✐③ t❛❧ q✉❡ AC =CA=I✱ ❡♥tã♦ C =CI =C(AB) = (CA)B = IB =B✳ ❆ss✐♠✱ s❡ A é ✐♥✈❡rsí✈❡❧✱ ♥♦t❛r❡♠♦s ♣♦rA−1

❛ ú♥✐❝❛ ♠❛tr✐③ t❛❧ q✉❡ A A−1

=A−1

A=I✳ ❉❡ss❛ ✉♥✐❝✐❞❛❞❡✱ s❡❣✉❡ q✉❡✱ s❡ Aé ✐♥✈❡rsí✈❡❧✱ ❡♥tã♦(A−1

)−1 =A✳ ❊①❡♠♣❧♦ ✶✳✶✾

✭✶✮ ❆ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡I é ✐♥✈❡rsí✈❡❧✱ ♣♦✐sI·I =I✱ ❞♦♥❞❡ I−1 =I✳

✭✷✮ ❆ ♠❛tr✐③ ♥✉❧❛ ✭q✉❛❞r❛❞❛✮O ♥ã♦ é ✐♥✈❡rsí✈❡❧ ♣♦✐s✱ ♣❛r❛ ❝❛❞❛ ♠❛tr✐③B✱ t❡♠♦sO·B =O 6=I✳ ✭✸✮ ✵ s✐♠♣❧❡s ❢❛t♦ ❞❡ ✉♠❛ ♠❛tr✐③ As❡r ♥ã♦ ♥✉❧❛✱ ♥ã♦ ❣❛r❛♥t❡✱ ❡♠ ❣❡r❛❧✱ q✉❡A s❡❥❛ ✐♥✈❡rsí✈❡❧✳ ❇❛st❛ ❝♦♥s✐❞❡r❛r ♦ ❡①❡♠♣❧♦

" 0 0 1 0

#

❡♠ q✉❡ "

0 0 1 0

# " a b c d

# =

" 0 0 a b

#

6=I ✳ ◆❛ r❡❛❧✐❞❛❞❡✱ ❞❡❝✐❞✐r s❡ ✉♠❛ ♠❛tr✐③ é ✐♥✈❡rsí✈❡❧ ❡ ❡①✐❜✐r s✉❛ ✐♥✈❡rs❛ é ✉♠❛ ♣♦t❡♥t❡ ❛♣❧✐❝❛çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ●❛✉ss✲❏♦r❞❛♥✳ ✭✹✮ ❙❡Aé ✉♠❛ ♠❛tr✐③ ✐♥✈❡rsí✈❡❧✱ ❡♥tã♦ s✉❛ tr❛♥s♣♦st❛tA t❛♠❜❡♠ é✱ ❝♦♠(t(A))−1

=t(A−1 )✳ ❉❡ ❢❛t♦✱ s❡A·A−1

=A−1

·A =I ❡♥tã♦ t(A·A−1

) =t(A−1

·A) =tI =I✱ ❞♦♥❞❡ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✱ t❡♥❞♦ ❡♠ ❝♦♥t❛ ❛ tr❛♥s♣♦s✐çã♦ ❞❡ ✉♠ ♣r♦❞✉t♦ ❞❡ ♠❛tr✐③❡s✱ ✐st♦ é✱t_(A−1

)·tA=tA·t(A−1 ) =I✳ ❱❡r✐✜q✉❡ ❛ ✈❛❧✐❞❛❞❡ ❞♦ r❡s✉❧t❛❞♦ r❡❝í♣r♦❝♦✿ t_{A ✐♥✈❡rsí✈❡❧=⇒ A ✐♥✈❡rsí✈❡❧✳}

✭✺✮ ❙❡❥❛♠A❡B♠❛tr✐③❡s q✉❛❞r❛❞❛s ❞❡ ♠❡s♠❛ ♦r❞❡♠✳ ❙❡A❡Bsã♦ ✐♥✈❡rsí✈❡✐s✱ ❡♥tã♦ ♦ ♣r♦❞✉t♦ABé ✐♥✈❡rsí✈❡❧ ❡ ✈❛❧❡(AB)−1

=B−1 ·A−1

❬♦❜s❡r✈❡ ❛ ♦r❞❡♠ ❞♦s ❢❛t♦r❡s❪✳ ❉❡ ❢❛t♦✱ t❡♠♦s(AB)(B−1 A−1

) = A(BB−1

)A−1

=AA−1

=I❀ ❛♥❛❧♦❣❛♠❡♥t❡✱(B−1 ·A−1

)(AB) =I✱ ❞♦♥❞❡ ❛ ❝♦♥❝❧✉sã♦✳ ◆❛ r❡❛❧✐❞❛❞❡✱ ❛s ♠❛tr✐③❡sA❡ B sã♦ ✐♥✈❡rsí✈❡✐s⇐⇒ AB é ✐♥✈❡rsí✈❡❧✳

✭✻✮ ❙❡❥❛A✐♥✈❡rsí✈❡❧✳ ❙❡ AB=AC✱ ❡♥tã♦ B =C ❬❝❢✳ ❊①❡♠♣❧♦ ✶✳✶✼✭✷✮ ❪✳ ❉❡ ❢❛t♦✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ ❞❛❞❛✱ à ❡sq✉❡r❞❛✱ ♣♦r A−1

✱ s❡❣✉❡✿ A−1

(AB) =A−1

(AC)✱ ❞♦♥❞❡ (A−1

A)B = (A−1

A)C✱ ♦✉ IB=IC ❡ B=C✳

✭✼✮ ❱♦❧t❛♥❞♦ ❛♦s s✐st❡♠❛s✱ s❡Aé ✉♠❛ ♠❛tr✐③ ✐♥✈❡rsí✈❡❧✱ ❡♥tã♦ ♦ s✐st❡♠❛ ❤♦♠♦❣ê♥❡♦AX= 0só ♣♦ss✉✐ ❛ s♦❧✉çã♦ tr✐✈✐❛❧X= 0✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ♦ s✐st❡♠❛ AX =Y ♣♦ss✉✐ ❛ ú♥✐❝❛ s♦❧✉çã♦ X=A−1

✶✹ ■♥tr♦❞✉çã♦ à ➪❧❣❡❜r❛ ▲✐♥❡❛r ✕ ❬❆♥t♦♥✐♦ ❈❛r❧♦s ✫ ❆♥❛ P❛✉❧❛ ▼❛rq✉❡s❪

❆t✐✈✐❞❛❞❡✲♣r♦♣♦st❛ ✶✳✷✵

✭✶✮ ❱❡r✐✜q✉❡ q✉❡ "

1 1 0 1

#−1 = " 1 −1 0 1 # ; " 2 1 1 1

#−1 = " 1 −1 −1 2 # ✳

✭✷✮ ❯♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A é ✐❞❡♠♣♦t❡♥t❡ s❡ A2 = A✳ ❈♦♥s✐❞❡r❛♥❞♦ ✉♠❛ t❛❧ ♠❛tr✐③✱ s❡❥❛♠ B = 2A−I ❡ C=I−A✳

✭❛✮ ❱❡r✐✜q✉❡ q✉❡ B2 =I ❡C2 =C✱ ❝♦♠AC =CA= 0❀

✭❜✮ ❈♦♥❝❧✉❛ q✉❡ ❛ ♠❛tr✐③ B é ✐♥✈❡rsí✈❡❧❀ ♣♦r ♦✉tr♦ ❧❛❞♦✱ s❡ A6=I✱ ❡♥tã♦ A♥ã♦ é ✐♥✈❡rsí✈❡❧✳ ✭✸✮ ❊♥❝♦♥tr❡ t♦❞❛s ❛s ♠❛tr✐③❡s A2×2 t❛✐s q✉❡A2 =I✳

✭✹✮ ▼♦str❡ q✉❡✱ s❡ ✉♠❛ ❧✐♥❤❛ ✭♦✉ ❝♦❧✉♥❛✮ ❞❡ ✉♠❛ ♠❛tr✐③A é ♥✉❧❛✱ ❡♥tã♦A ♥ã♦ é ✐♥✈❡rsí✈❡❧✳

✶✳✷✶ ❖ ▼➱❚❖❉❖ ❉❊ ●❆❯❙❙✲❏❖❘❉❆◆

❱✐♠♦s ♥♦ ➓✶✳✷ q✉❡ ♦ ❡s❝❛❧♦♥❛♠❡♥t♦ ❞❡ ✉♠ s✐st❡♠❛ ❝♦rr❡s♣♦♥❞❡ ❛ r❡❛❧✐③❛r ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s s♦❜r❡ ❛s ❧✐♥❤❛s ❞❛ ♠❛tr✐③ ✭❛✉♠❡♥t❛❞❛✮ q✉❡ ♦ r❡♣r❡s❡♥t❛✳ ◆❛ r❡❛❧✐❞❛❞❡✱ ✈❛♠♦s ❡①✐❣✐r ❛❧❣♦ ♠❛✐s✿ ♥♦ss♦ s✐st❡♠❛ ✭♦✉ s✉❛ ♠❛tr✐③✮ ❛❧é♠ ❞❡ ❡s❝❛❧♦♥❛❞♦✱ ❞❡✈❡rá ❡st❛r s♦❜ ❢♦r♠❛ r❡❞✉③✐❞❛✳

❯♠ s✐st❡♠❛ é ❡s❝❛❧♦♥❛❞♦ r❡❞✉③✐❞♦ q✉❛♥❞♦✱ ❛❧é♠ ❞❡ ❡s❝❛❧♦♥❛❞♦✱ ♦s ❝♦❡✜❝✐❡♥t❡s ❧í❞❡r❡s sã♦ ✐❣✉❛✐s ❛ ✉♠ ❡ ♦s ❞❡♠❛✐s ❡❧❡♠❡♥t♦s ❞❛ ❝♦❧✉♥❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ❧✐❞❡r sã♦ t♦❞♦s ♥✉❧♦s✳ ❈♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❧✐♥❤❛s ♥✉❧❛s ✭s❡ ❤♦✉✈❡r✮ ♦❝♦rr❡♠ ❛♣ós t♦❞❛s ❛s ❧✐♥❤❛s ♥ã♦ ♥✉❧❛s✳

❖❜s❡r✈❡♠♦s q✉❡ tr❛♥s❢♦r♠❛r ✉♠ s✐st❡♠❛ ✭♦✉ s✉❛ ♠❛tr✐③✮ ❞❛ ❢♦r♠❛ ❡s❝❛❧♦♥❛❞❛ ♣❛r❛ ❛ ❢♦r♠❛ r❡❞✉③✐❞❛ ♥ã♦ ♦❢❡r❡❝❡ ❣r❛♥❞❡s ❞✐✜❝✉❧❞❛❞❡s✿ ❜❛st❛ ❞✐✈✐❞✐r ❛ ❡q✉❛çã♦ ♣❡❧♦ ❡❧❡♠❡♥t♦ ❧í❞❡r✱ ♦ q✉❡ ♦ ❢❛rá ✉♥✐tár✐♦✱ ❡✱ ♣♦r ♠ú❧t✐♣❧♦s ❝♦♥✈❡♥✐❡♥t❡s ❞❡ss❛ ❧✐♥❤❛✱ ❛♥✉❧❛r ♦s ❞❡♠❛✐s ❡❧❡♠❡♥t♦s ❞❛ ❝♦❧✉♥❛ ❞♦ ❧í❞❡r✳

❊①❡♠♣❧♦ ✶✳✷✶

✭✶✮ ❘❡t♦♠❛♥❞♦ ♦ ❊①❡♠♣❧♦ ✹

ˆ

A=



 

1 1 0 0

2 −1 3 3 1 −2 −1 3



 

−2L1+L2 −−−−−−→

−L1+L3 

 

1 1 0 0

0 −3 3 3 0 −3 −1 3



 

−L2+L3 −−−−−→



 

1 1 0 0

0 −3 3 3 0 0 −4 0



 

−(1/3)L2 −−−−−−→

−(1/4)L3 

 

1 1 0 0

0 1 −1 −1

0 0 1 0



 

−L2+L1 −−−−−→ 

 

1 0 1 1

0 1 −1 −1

0 0 1 0



 

L3+L2 −−−−−→

−L3+L1 

 

1 0 0 1 0 1 0 −1 0 0 1 0





✳ ❙❡❣✉❡ ❛ ú♥✐❝❛ s♦❧✉çã♦X= (x, y, z) = (1,−1,0)✳

✭✷✮ ❖✉tr❛ r❡❞✉çã♦ ❡s❝❛❧♦♥❛❞❛ ✭✐❞❡♥t✐✜q✉❡ ❛s ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s✦✮✿

ˆ A=



 

1 1 0 1 2 1 2 1 1 2 1 2

  −→   

1 1 1 1 0 −1 0 −1 0 1 0 1

  −→   

1 0 1 0 0 1 0 1 0 0 0 0





✳ ❊♥✜♠✱ X= (x, y, z) = (−z,1, z) z∈R✳

✭✸✮ ❆s ♣♦ssí✈❡✐s ♠❛tr✐③❡s2×2 ❡s❝❛❧♦♥❛❞❛s r❡❞✉③✐❞❛s sã♦ ❞❛s ❢♦r♠❛s ❛❜❛✐①♦✿ " 0 0 0 0 # ❀ " 1 a 0 0 #

✱ ♦♥❞❡ a∈R❀ " 0 1 0 0 # ❀ " 1 0 0 1 # ✳

✶✳✷✷ ▼❛tr✐③❡s ❡❧❡♠❡♥t❛r❡s

❙✐st❡♠❛s ❧✐♥❡❛r❡s ❡ ♠❛tr✐③❡s ✶✺

P♦r ❡①❡♠♣❧♦✱ sã♦ ❡❧❡♠❡♥t❛r❡s ❛s ♠❛tr✐③❡sE1✱E2 ❡E3 ❛❜❛✐①♦✳ ❆ ♣r✐♠❡✐r❛ é ♦❜t✐❞❛ ❞❡I ♠✉❧t✐♣❧✐❝❛♥❞♦

♣♦r ✹ ❛ s❡❣✉♥❞❛ ❧✐♥❤❛❀ ❛ ♠❛tr✐③ E2 é ♦❜t✐❞❛ s♦♠❛♥❞♦ à t❡r❝❡✐r❛ ❧✐♥❤❛ ❞❡ I ❛ s✉❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛

♠✉❧t✐♣❧✐❝❛❞❛ ♣♦r ✺✳ ❈♦♠♦ ❢♦✐ ♦❜t✐❞❛E3❄

E1=



 

1 0 0 0 4 0 0 0 1



 ✱E2 =



 

1 0 0 0 1 0 5 0 1



 ✱E3 =



 

0 0 1 0 1 0 1 0 0



 ✳

✶✳✷✸ Pr♦♣r✐❡❞❛❞❡s ❞❛s ♠❛tr✐③❡s ❡❧❡♠❡♥t❛r❡s

✭❛✮ ❈♦♥s✐❞❡r❡♠♦s ❛ ♠❛tr✐③ ❡❧❡♠❡♥t❛r E = e(I)✳ ❙❡❥❛ e(A) ❛ ♠❛tr✐③ ♦❜t✐❞❛ ❞❡ ✉♠❛ ♠❛tr✐③ A ♣❡❧❛ ❛♣❧✐❝❛çã♦ ❞❛ ♦♣❡r❛çã♦ ❡❧❡♠❡♥t❛re✳ ❊♥tã♦✱e(A) =EA✳

P♦r ❡①❡♠♣❧♦✱ s❡ eé ❛ ♦♣❡r❛çã♦ q✉❡ ♣❡r♠✉t❛ ❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ❝♦♠ ❛ t❡r❝❡✐r❛✱ t❡♠♦s✿

E=e(I) = 

 

1 0 0 0 0 1 0 1 0



 ✱A=



 

a1 b1 c1

a2 b2 c2

a3 b3 c3





✱EA= 

 

1 0 0 0 0 1 0 1 0

  ·   

a1 b1 c1

a2 b2 c2

a3 b3 c3

  =   

a1 b1 c1

a3 b3 c3

a2 b2 c2





=e(A)✳

✭❜✮ ❚♦❞❛ ♠❛tr✐③ ❡❧❡♠❡♥t❛r é ✐♥✈❡rsí✈❡❧✳

❉❡ ❢❛t♦✱ s❡e é ✉♠❛ ♦♣❡r❛çã♦ ❡❧❡♠❡♥t❛r✱ s❡❥❛ e1 ❛ ♦♣❡r❛çã♦ ❡❧❡♠❡♥t❛r q✉❡ r❡✈❡rt❡e ❬❝❢✳ Pr♦♣✳ ✶✳✷❪✱

✐st♦ é✱ s❡E =e(I) ❡ E1 =e1(I)✱ ❡♥tã♦ e1(E) =e1(e(I)) =I✱ ✐st♦ é✱ E1·E =I✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱

E·E1=I✳

P♦r ❡①❡♠♣❧♦✱ é ❢á❝✐❧ ✈❡r✐✜❝❛r ❛s s❡❣✉✐♥t❡s ✐♥✈❡rs❛s ❞❡ ♠❛tr✐③❡s ❡❧❡♠❡♥t❛r❡s✿ 

 

1 0 0 0 0 1 0 1 0

   −1 =   

1 0 0 0 0 1 0 1 0

  ❀   

1 0 0 0 4 0 0 0 1

   −1 =   

1 0 0 0 1₄ 0 0 0 1

  ❀   

1 0 0 0 1 0 5 0 1

   −1 =   

1 0 0 0 1 0 −5 0 1



 ✳

✶✳✷✹ ▼❛tr✐③❡s ❡q✉✐✈❛❧❡♥t❡s ♣♦r ❧✐♥❤❛s

◆♦ ♠❡s♠♦ ❝♦♥t❡①t♦ ❞❡ ❬✶✳✸❪✱ ❞✐r❡♠♦s q✉❡ ❞✉❛s ♠❛tr✐③❡s ❞❡ ♠❡s♠❛ ♦r❞❡♠A❡B✭♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ q✉❛❞r❛❞❛s✮ sã♦ ❡q✉✐✈❛❧❡♥t❡s ♣♦r ❧✐♥❤❛s✱ ❡ ♥♦t❛r❡♠♦sA∼B✱ s❡ ❛ ♠❛tr✐③B é ♦❜t✐❞❛ ❞❡ A♣♦r ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s s♦❜r❡ ❛s ❧✐♥❤❛s ❞❡ A✳ ❖❜s❡r✈❡♠♦s q✉❡ ❛ r❡❧❛çã♦ A ∼ B é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ ✐st♦ é✱ ✈❛❧❡♠ ❛s ♣r♦♣r✐❡❞❛❞❡s r❡✢❡①✐✈❛ ✭A∼A✮✱ s✐♠étr✐❝❛ ✭A ∼B =⇒ B∼A✮ ❡ tr❛♥s✐t✐✈❛ ✭A∼B✱B ∼C =⇒ A∼C✮✳

❚❡♥❞♦ ❡♠ ❝♦♥t❛ ❛ Pr♦♣r✐❡❞❛❞❡ ✶✳✷✸✭❛✮✱ s❡A∼B✱ ❡♥tã♦ ❡①✐st❡♠ ♠❛tr✐③❡s ❡❧❡♠❡♥t❛r❡s E1, E2, . . . , Et t❛✐s q✉❡Et·Et−1·. . .·E2·E1·A=B✳ P♦♥❞♦P =Et·Et−1·. . .·E2·E1·I✱ ✈❡♠♦s q✉❡B =P A✳ ❆❧é♠ ❞✐ss♦✱ ❛ ♠❡s♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s q✉❡ tr❛♥s❢♦r♠❛ A ❡♠ B✱ t❛♠❜é♠ tr❛♥s❢♦r♠❛ I ❡♠P✿ [A|I] −ei→ [B |P]✳ ❘❡s✉♠✐♥❞♦ ✉♠ ♣♦✉❝♦✱ ❞❡st❛q✉❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❢✉♥❞❛♠❡♥t❛❧✿

✶✳✷✺ Pr♦♣r✐❡❞❛❞❡ ❝❛r❛❝t❡ríst✐❝❛ ❞❛ ❡q✉✐✈❛❧ê♥❝✐❛ ♣♦r ❧✐♥❤❛s

❙❡❥❛♠ A ❡ B ♠❛tr✐③❡s m×n❀ A ∼ B ⇐⇒ B = P A✱ ♦♥❞❡ P é ✉♠ ♣r♦❞✉t♦ ❞❡ ♠❛tr✐③❡s m×m ❡❧❡♠❡♥t❛r❡s✳

❊①❡♠♣❧♦ ✶✳✷✻

❈♦♥s✐❞❡r❡♠♦s ❛ s❡q✉ê♥❝✐❛ ❞❡ ❡s❝❛❧♦♥❛♠❡♥t♦ ❡ r❡❞✉çã♦ ♣♦r ❧✐♥❤❛s✿

(A|I) = 

 

1 1 1 1 0 0 1 2 2 0 1 0 2 3 3 0 0 1

   −→   

1 1 1 1 0 0 0 1 1 −1 1 0 0 1 1 −2 0 1

  −→   

1 0 0 2 −1 0 0 1 1 −1 1 0 0 0 0 −1 −1 1





= (B|P)✳

✶✻ ■♥tr♦❞✉çã♦ à ➪❧❣❡❜r❛ ▲✐♥❡❛r ✕ ❬❆♥t♦♥✐♦ ❈❛r❧♦s ✫ ❆♥❛ P❛✉❧❛ ▼❛rq✉❡s❪

❊♠ s❡❣✉✐❞❛✱ ♣♦❞❡♠♦s ❡①❛♠✐♥❛r ✉♠❛ ❛♣❧✐❝❛çã♦ ❡s♣❡❝✐❛❧ ❡♠ q✉❡ ✉♠ s✐st❡♠❛ ❤♦♠♦❣ê♥❡♦ ♣♦ss✉✐ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧✳

✶✳✷✼ ❆♣❧✐❝❛çã♦

❙❡❥❛ Am×n✱ ❝♦♠m < n✳ ❊♥tã♦✱ ♦ s✐st❡♠❛ ❤♦♠♦❣ê♥❡♦ AX= 0 ♣♦ss✉✐ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧✳

❉❡ ❢❛t♦✱ ❡s❝❛❧♦♥❛♥❞♦ ❡ r❡❞✉③✐♥❞♦ ♦ s✐st❡♠❛ AX = 0✱ ♦❜t❡♠♦s ♦ s✐st❡♠❛ ❡q✉✐✈❛❧❡♥t❡ RX = 0✳ ❙❡ r é ♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ♥ã♦ ♥✉❧❛s ❞❛ ♠❛tr✐③ R✱ t❡♠♦s r ≤m < n❀ ❧♦❣♦✱ ❤á n−r ✈❛r✐á✈❡✐s ❧✐✈r❡s✱ q✉❡ ❢♦r♥❡❝❡♠ ❛❧❣✉♠❛ s♦❧✉çã♦ ✐❣✉❛❧ ❛ ✶ ✭♥ã♦ ♥✉❧❛✮ ❬r❡✈❡r ❛ ❞✐s❝✉ssã♦ ❞♦ ❘❡s✉♠♦ ✶✳✼✭✷✮❪✳

Pr❛t✐q✉❡ ✉♠ ♣♦✉❝♦

❖ r❡s✉❧t❛❞♦ ❛❝✐♠❛ ❣❛r❛♥t❡ s♦❧✉çã♦ X 6= O ♥♦ ❝❛s♦ ❝♦♥s✐❞❡r❛❞♦✱ ♦✉ s❡❥❛✱ ♠❡♥♦s ❡q✉❛çõ❡s ❞♦ q✉❡ ✐♥❝ó❣♥✐t❛s✳ ❉ê ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ♠❛tr✐③ 2×2 ❡ ❞❡ ✉♠❛ ♠❛tr✐③ 3×2 ❝✉❥♦s ♥ú❝❧❡♦s sã♦ tr✐✈✐❛✐s✳

✶✳✷✽ ❈♦r♦❧ár✐♦

❙❡❥❛ A ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛n×n✳ ❚❡♠♦s✿ A∼I ⇐⇒ ♦ s✐st❡♠❛ AX = 0 ♣♦ss✉✐ ❛♣❡♥❛s ❛ s♦❧✉çã♦ tr✐✈✐❛❧✳

❙❡ A ∼ I✱ ❡♥tã♦ ♦s s✐st❡♠❛s AX = 0 ❡ IX = 0 sã♦ ❡q✉✐✈❛❧❡♥t❡s✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ R ∼ A ❡s❝❛❧♦♥❛❞❛ r❡❞✉③✐❞❛✱ ❝♦♠ r ♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ♥ã♦ ♥✉❧❛s ❞❡ R✳ ❈♦♠♦RX = 0 ♥ã♦ ❛❞♠✐t❡ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧✱ ✈❡♠r≥n❀ ♣♦r ♦✉tr♦ ❧❛❞♦✱R♣♦ss✉✐n❧✐♥❤❛s✱ ♦✉ s❡❥❛r =n✳ ❖r❛✱ ❡♥tã♦R♣♦ss✉✐ ❡❧❡♠❡♥t♦ ❧✐❞❡r ✐❣✉❛❧ ❛ ✶ ❡♠ ❝❛❞❛ ✉♠❛ ❞❡ s✉❛sn ❧✐♥❤❛s✱ ✐st♦ éR=I✳

✶✳✷✾ ❈➪▲❈❯▲❖ ❉❆ ▼❆❚❘■❩ ■◆❱❊❘❙❆

❚❡♦r❡♠❛✳ ❯♠❛ ♠❛tr✐③A é ✐♥✈❡rsí✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱A∼I✳ ◆❡ss❡ ❝❛s♦✱ ❛ s❡q✉ê♥❝✐❛ ❞❡ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s q✉❡ tr❛♥s❢♦r♠❛A ❡♠I✱ t❛♠❜é♠ tr❛♥s❢♦r♠❛I ❡♠A−1✳

❈♦♠ ❡❢❡✐t♦✱ ✈✐♠♦s ❡♠ ❬✶✳✷✺❪ q✉❡✱ s❡ A ∼ B✱ ❡♥tã♦ B = P A✱ ♦♥❞❡ P é ✉♠ ♣r♦❞✉t♦ ❞❡ ♠❛tr✐③❡s ❡❧❡♠❡♥t❛r❡s✱ ♣♦rt❛♥t♦ ✐♥✈❡rsí✈❡❧✳ ▲♦❣♦✱ s❡ A ∼I✱ ❡♥tã♦ I = P A✱ ❝♦♠ P = Et·. . .·E2·E1✳ ❙❡❣✉❡

q✉❡ A=P−1

·I é ❜❡♠ ✐♥✈❡rsí✈❡❧ ❡ A =E−1

1 ·E

−1

2 ·. . .·E

−1

t ·I✳ ❊♥✜♠✱ A −1

=Et·. . .·E2·E1·I✱

♦ q✉❡ ❣❛r❛♥t❡ ❛ ú❧t✐♠❛ ❛✜r♠❛çã♦ ❞♦ t❡♦r❡♠❛✱ ✐st♦ é✱ [A |I] −ei→ [I |A−1

]✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ A é ✐♥✈❡rsí✈❡❧✱ ❡♥tã♦AX = 0 só ♣♦ss✉✐ ❛ s♦❧✉çã♦ tr✐✈✐❛❧✳ ❙❡❣✉❡✱ ❡♥tã♦✱ ❞♦ ❈♦r♦❧ár✐♦ ✶✳✷✽✱ q✉❡A∼I✳

✶✳✸✵ ❈♦r♦❧ár✐♦

❙❡❥❛♠A ❡ B ♠❛tr✐③❡s m×n✳ ❊♥tã♦A∼B ⇐⇒B =P A✱ ♦♥❞❡ P é ✉♠❛ ♠❛tr✐③ m×m ✐♥✈❡rsí✈❡❧✳

❊①❡♠♣❧♦ ✶✳✸✶

❱❡r✐✜❝❛r s❡ ❛ ♠❛tr✐③ ❞❛❞❛ é ✐♥✈❡rsí✈❡❧ ❡ ❛❝❤❛r✱ q✉❛♥❞♦ ♣♦ssí✈❡❧✱ s✉❛ ✐♥✈❡rs❛✱ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ r❡❞✉çã♦✲❡s❝❛❧♦♥❛❞❛[A|I] −ei→ [I |A−1]✳

[A|I] =



 

1 0 3 1 0 0 0 1 4 0 1 0

−1 1 2 0 0 1



 −→



 

1 0 3 1 0 0 0 1 4 0 1 0 0 1 5 1 0 1



 −→



 

1 0 3 1 0 0 0 1 4 0 1 0 0 0 1 1 −1 1



 −→



 

1 0 0 −2 3 −3 0 1 0 −4 5 −4 0 0 1 1 −1 1



 

❙❡❣✉❡ ❞❡ ✶✳✸✵ q✉❡A é ✐♥✈❡rsí✈❡❧ ❡A−1 =



 

−2 3 −3 −4 5 −4 1 −1 1



 ✳

❙✐st❡♠❛s ❧✐♥❡❛r❡s ❡ ♠❛tr✐③❡s ✶✼

❆t✐✈✐❞❛❞❡✲♣r♦♣♦st❛ ✶✳✸✷

❯s❡ ♦ ♠ét♦❞♦ ❞❡ r❡❞✉çã♦✲❡s❝❛❧♦♥❛❞❛[A|I] −ei→ [I |A−1

]♣❛r❛ ❞❡❝✐❞✐r s❡ ❛ ♠❛tr✐③Aé ✐♥✈❡rsí✈❡❧❀ ❡♠ ❝❛s♦ ❛✜r♠❛t✐✈♦✱ ❡①✐❜✐r s✉❛ ✐♥✈❡rs❛✳

✭✶✮A= 

 

1 1 3 0 2 4 −1 1 2





❀ ✭✷✮A= 

 

1 2 6 0 1 5 2 3 7



 ✳

❊♥❝❡rr❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ♠❛tr✐③❡s ✐♥✈❡rsí✈❡✐s✱ ✈❡r✐✜q✉❡♠♦s ❛❜❛✐①♦ ✉♠ r❡s✉❧t❛❞♦ ♣r♦♠❡t✐❞♦✳ ✶✳✸✸ ▼❛tr✐③❡s ✐♥✈❡rsí✈❡✐s à ❡sq✉❡r❞❛ ❡ à ❞✐r❡✐t❛

❯♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❝♦♠ ✐♥✈❡rs❛ à ❡sq✉❡r❞❛ ♦✉ à ❞✐r❡✐t❛ é ✐♥✈❡rsí✈❡❧✳

❉❡ ❢❛t♦✱ s❡Bé ✉♠❛ ♠❛tr✐③ ✐♥✈❡rs❛ à ❡sq✉❡r❞❛ ❞❛ ♠❛tr✐③A✱ ❡♥tã♦BA=I✱ ❞♦♥❞❡ ♦ s✐st❡♠❛ ❤♦♠♦❣ê♥❡♦ AX = 0 ♣♦ss✉✐ ❛♣❡♥❛s ❛ s♦❧✉çã♦ tr✐✈✐❛❧✱ ♣♦✐s B(AX) = B0 =⇒ (BA)X = 0 ♦✉ IX = 0 ❡ X = 0✳ ❙❡❣✉❡ q✉❡ A é ✐♥✈❡rsí✈❡❧✱ B = A−1

❡ B−1

= A✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ C é ✉♠❛ ✐♥✈❡rs❛ à ❞✐r❡✐t❛ ❞❡ A✱ ❡♥tã♦ AC =I✱ ❞♦♥❞❡ C ♣♦ss✉✐ ✉♠❛ ✐♥✈❡rs❛ à ❡sq✉❡r❞❛ ❡✱ ♣♦rt❛♥t♦✱ C é ✐♥✈❡rsí✈❡❧✳ ❊♥✜♠✱ ❝♦♠♦ A=C−1

✱ ✈❡♠ q✉❡A é ✐♥✈❡rsí✈❡❧ ❡A−1 =C✳

✶✳✸✹ ❙✐st❡♠❛s ❞❡ ❈r❛♠❡r

❯♠ s✐st❡♠❛ ❧✐♥❡❛rAX =Y é ✉♠ s✐st❡♠❛ ❞❡ ❈r❛♠❡r s❡A é ✐♥✈❡rsí✈❡❧✳ P♦rt❛♥t♦✱ ❡ss❡ s✐st❡♠❛ ♣♦ss✉✐ ❛♣❡♥❛s ✉♠❛ s♦❧✉çã♦X=A−1

Y✳

❊①❡♠♣❧♦ ✶✳✸✺

◆♦ ❡①❡♠♣❧♦ ✶✳✸✶✱ ❝♦♠♦ ❛ ♠❛tr✐③A é ✐♥✈❡rsí✈❡❧✱ ❡♥tã♦AX=Y é ✉♠ s✐st❡♠❛ ❞❡ ❈r❛♠❡r✳

❉❛❞♦Y = 

 

1 1 3





✱ t❡♠♦sX=A −1

Y = 

 

−2 3 −3 −4 5 −4 1 −1 1



 ·



 

1 1 3



 =



 

−8 −11 3



 ✳

❖✉tr♦ ♠ét♦❞♦ ♣♦❞❡r✐❛ s❡r ✉s❛❞♦✱ ♦ ❞❡ r❡❛❧✐③❛r ✉♠ ❡s❝❛❧♦♥❛♠❡♥t♦ ♠❛✐s ❛♠♣❧✐❛❞♦✱ ❡ ♦❜t❡♥❞♦ ♠❛✐♦r ♥ú♠❡r♦ ❞❡ ✐♥❢♦r♠❛çõ❡s r❡❧❡✈❛♥t❡s

[A|Y |I]→[I|X|A−1 ]✳

[A|Y|I] = 

 

1 0 3 1 1 0 0 0 1 4 1 0 1 0 −1 1 2 3 0 0 1



  →



 

1 0 0 −8 −2 3 −3 0 1 0 −11 −4 5 −4 0 0 1 3 1 −1 1





= [I|X|A −1

]

Pr❛t✐q✉❡ ✉♠ ♣♦✉❝♦✳

✶✽ ■♥tr♦❞✉çã♦ à ➪❧❣❡❜r❛ ▲✐♥❡❛r ✕ ❬❆♥t♦♥✐♦ ❈❛r❧♦s ✫ ❆♥❛ P❛✉❧❛ ▼❛rq✉❡s❪

❘❊❙P❖❙❚❆❙ ❉❆❙ ❆❚■❱■❉❆❉❊❙ P❘❖P❖❙❚❆❙ ✶✳✻ ✭✶✮

✭✐✮ (

2x+ 3y= 13 x− y=−1 →

(

x− y=−1 2x+ 3y= 13 →

(

x− y=−1 5y= 15 ✳ ▲♦❣♦✱ y= 3✱x= 2 ❡ X= (2,3)✳

✭✐✐✮     

x+ 3y= 1 2x+ y=−3 x+ y= 0

→     

x+ 3y= 1 −5y=−5 −2y=−1

→     

x+ 3y= 1 y= 1 y= 1/2

✳ ❙✐st❡♠❛ ✐♥❝♦♠♣❛tí✈❡❧✳ ✭✐✐✐✮     

x+ 2y = 4 y− z= 0 x + 2z= 4

→     

x+ 2y = 4 y− z= 0 −2y+ 2z= 0

→     

x+ 2y = 4 y−z= 0 0 = 0 ❙✐st❡♠❛ ✐♥❞❡t❡r♠✐♥❛❞♦ X= (4−2z, z, z)✱z∈R✳

✭✐✈✮     

4x+ y−z= 1 2x+ 2y+z= 5 x− y−z=−4

✳ ❙♦❧✉çã♦ ú♥✐❝❛✿ X= (−1,4,−1)✳ ❱❡r✐✜q✉❡ ❝♦♠ ✉♠ ❡s❝❛❧♦♥❛♠❡♥t♦✳

✶✳✻ ✭✷✮ ❊s❝❛❧♦♥❛♥❞♦ ♦ s✐st❡♠❛ ❞❛❞♦✱ ♦❜t❡♠♦s✿ (

x− y= 1 3x−3y=α →

(

x−y= 1

0 =α−3 ❆ss✐♠✱ s❡ α6= 3✱ ❡♥tã♦ ♦ s✐st❡♠❛ é ✐♠♣♦ssí✈❡❧✳ ❙❡α= 3✱ ♦ s✐st❡♠❛ é ♣♦ssí✈❡❧ ✐♥❞❡t❡r♠✐♥❛❞♦✱ ❝♦♠ s♦❧✉çõ❡s X= (x, y) = (1 +y, y)✱y∈R✳

✶✳✻ ✭✸✮ ❆s ❝♦♥❞✐çõ❡s ❞♦ ♣r♦❜❧❡♠❛✱ f(1) = 2✱f(−1) = 6❡f(2) = 3✱ ❢♦r♥❡❝❡♠ ♦ s✐st❡♠❛✿ 

 

 

a+ b+c= 2 a− b+c= 6 4a+ 2b+c= 3

→     

a+ b+ c= 2 −2b = 4 −2b−3c=−5

→     

a+ b+ c= 2 −2b = 4 −3c=−9 ✳

❙❡❣✉❡✱ ❡♥tã♦✿ c= 3✱b=−2 ❡a= 1✱ ❞♦♥❞❡ ♦ tr✐♥ô♠✐♦f(x) =x2−2x+ 3✳

✶✳✾ ❊①❡♠♣❧✐✜q✉❡♠♦s ❝♦♠ ♦ ♣r♦❜✳ ✶✳✻ ✭✭✶✮✐✈✮✳

ˆ

A=



 

4 1 −1 1

2 2 1 5

1 −1 −1 −4



 

L1↔L3 −−−−−→



 

1 −1 −1 −4

2 2 1 5

4 1 −1 1



 

−2L1+L2 −−−−−−→

−4L1+L3 

 

1 −1 −1 −4

0 4 3 13

0 5 3 17



 

−L2+L3 −−−−−→



 

1 −1 −1 −4

0 4 3 13

0 1 0 4



 

L2↔L3 −−−−−−→

−4L2+L3 

 

1 −1 −1 −4

0 1 0 4

0 0 3 −3





✳ ❊♥✜♠✱ ♦❜t❡♠♦s✿ z = −1✱ y = 4 ❡ x = −1✱ ✐st♦ é✱ s♦❧✉çã♦ ú♥✐❝❛ X =

(−1,4,−1)✳

✶✳✶✷ ✭✶✮ ❡ ✭✷✮✿ ❛♣❡♥❛s ❛ s♦❧✉çã♦ tr✐✈✐❛❧X= (0,0)❀ ✭✸✮X = (z,−z, z) =z(1,−1,1)✱z∈R✳ ✭✹✮ P❛r❛ ♣❡r♠✉t❛r✱ ♣♦r ❡①❡♠♣❧♦✱ ❛s ❧✐♥❤❛s L1 ❡L3✱ ♣♦❞❡♠♦s ♣r♦❝❡❞❡r ❝♦♠♦ ❛❜❛✐①♦✿

L1

L3

L1−L3 −−−−→

L3

L1−L3 −−−−→ L1 −L3 −−−→ L1 L3 L1

✶✳✶✽ ✭✶✮ ❚❡♠♦s✱ s✉❝❡ss✐✈❛♠❡♥t❡✱

2A+E=

"

3 5

−7 4

#

❀AB=

"

−3 7 1

−2 14 −4

#

❀BC=

"

2 −3 −1 2 9 −1

#

❀EA=

"

5 1 1 3

#

❀ DB=



 

4 −16 2

0 0 0

−3 3 3



 

✭✷✮❙❡❥❛X =

" a b c d #

✳ ❈♦♠♣❛r❛♥❞♦ ❛s ♠❛tr✐③❡sAX=XA✱ t❡♠♦sa=b+d❡c= 0✱ ❞♦♥❞❡X =

"

b+d b

0 d

#