Gabarito Lista 01

❙♦❧✉çõ❡s ❞♦s ❊①❡r❝í❝✐♦s ❞❡ ❆◆
▲✐st❛ ■ ✲ ❙✐st❡♠❛s ▲✐♥❡❛r❡s

✷✽ ❞❡ ▼❛rç♦ ❞❡ ✷✵✶✷
❊st❡s ♣r✐♠❡✐r♦s ❡①❡r❝í❝✐♦s ❞❡✈❡♠ s❡r ❢❡✐t♦s s❡♠ ✉s❛r ❛s ❢✉♥çõ❡s ❞♦ s❝✐❧❛❜
♣❛r❛ s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s✳ ❊❧❡s sã♦ ♣❡q✉❡♥♦s ❡ s✐♠♣❧❡s ♣❛r❛ q✉❡ ✈♦❝ê r❡s♦❧✈❛✲♦s ❞✐r❡t❛♠❡♥t❡ ❢❛③❡♥❞♦ ❛s ❝♦♥t❛s ❝♦♠ ♣❛♣❡❧ ❡ ❧á♣✐s✳
❈♦♥s✐❞❡r❡ ♦ s✐st❡♠❛ ❧✐♥❡❛r Ax = b ♦♥❞❡ ❆ t❡♠ m ❧✐♥❤❛s ❡ n
❝♦❧✉♥❛s✳ ❙✉❜st✐t✉❛ ??? ♣❡❧❛s r❡s♣♦st❛s ❝♦rr❡t❛s ♥❛s ❛✜r♠❛çõ❡s ❛❜❛✐①♦✳
❊①❡r❝í❝✐♦ ✵✳✶

• ❙❡ m < n✱ ♦ s✐st❡♠❛ é s✉❜❞❡t❡r♠✐♥❛❞♦ ❡ ❡♠ ❣❡r❛❧ ♣♦ss✉✐ ✐♥✜♥✐t❛s

s♦❧✉çõ❡s✳

• ❙❡ m = n✱ ♦ s✐st❡♠❛ ♣♦❞❡ t❡r ③❡r♦✱ ✉♠❛ ♦✉ ✐♥✜♥✐t❛s s♦❧✉çõ❡s✳
• ❙❡ m > n✱ ♦ s✐st❡♠❛ é s♦❜r❡❞❡t❡r♠✐♥❛❞♦ ❡✱ ❡♠ ❣❡r❛❧✱ ♥ã♦ t❡rá s♦❧✉çã♦✳
❊❧❡ t❡rá s♦❧✉çã♦ ❛♣❡♥❛s ♥♦ ❝❛s♦ ❞♦ ✈❡t♦r b s❡r ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡
❞❛s ❝♦❧✉♥❛s ❞❡ A✳

❘❡s♦❧✈❛ ♦ s✐st❡♠❛ ❧✐♥❡❛r ♣❡❧♦ ♠ét♦❞♦ ❞❡ ❡❧✐♠✐♥❛çã♦ ❞❡ ●❛✉ss
✭s❡♠ ✉s❛r ❛ ♣✐✈♦t❛çã♦ ♣❛r❝✐❛❧✮✳
❊①❡r❝í❝✐♦ ✵✳✷







x1 + 6x2 + 2x3 + 4x4 = 8
3x1 + 19x2 + 4x3 + 15x4 = 25

1x1 + 4x2 + 8x3 − 12x4 = 18



5x1 + 33x2 + 9x3 + 3x4 = 72

❈❤❡q✉❡ s❡ s✉❛ s♦❧✉çã♦ é ❡①❛t❛ ✈❡r✐✜❝❛♥❞♦ s❡ ❡❧❛ ❞❡ ❢❛t♦ s❛t✐s❢❛③ ♦ s✐st❡♠❛
❛❝✐♠❛✳ ❊s❝r❡✈❛ ♦ s✐st❡♠❛ ❛❝✐♠❛ ♥❛ ❢♦r♠❛ Ax = b✳ ❊s❝r❡✈❛✲♦ t❛♠❜é♠ ♥❛
❢♦r♠❛ ♠❛tr✐❝✐❛❧ ❛✉♠❡♥t❛❞❛✳ ❆ s❡❣✉✐r✱ ♦❜t❡♥❤❛ ♦ s✐st❡♠❛ ❡q✉✐✈❛❧❡♥t❡ U x = c
✶

r❡s✉❧t❛♥t❡ ❞❛ ❡❧✐♠✐♥❛çã♦ ❣❛✉ss✐❛♥❛✳ ❋✐♥❛❧♠❡♥t❡✱ ❡s❝r❡✈❛ ♦ s✐st❡♠❛ Ax = b
♥❛ ❢♦r♠❛ ❞❛ ❞❡❝♦♠♣♦s✐çã♦ LU x = b ✉s❛♥❞♦ ♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❛ ❡❧✐♠✐♥❛çã♦
❣❛✉ss✐❛♥❛✳
❙♦❧✉çã♦✿

❖ s✐st❡♠❛ Ax = b ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦






1 6 2 4
3 19 4 15
1 4 8 −12
5 33 9 3







x1 x2 x3 x4 








=





8
25
18
72







❡ ❡♠ s✉❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧ ❛✉♠❡♥t❛❞❛ ❝♦♠♦






1 6 2 4
8
3 19 4 15 25
1 4 8 −12 18
5 33 9 3 72




.


❘❡❛❧✐③❛♥❞♦ ❛ ❡❧✐♠✐♥❛çã♦ ❣❛✉ss✐❛♥❛ ❡♥❝♦♥tr❛♠♦s






1
0
0
0

6 2
4
8
1 −2 3
1
0 2 −10 12
0 0 −1 −1







♦✉ ❛✐♥❞❛






1
0
0
0

6 2
4
1 −2 3
0 2 −10
0 0 −1







x1 x2 x3 x4 








=





8
1
12
−1




.


▲♦❣♦✱ x4 x3 x2 x1

= 1,
= 6 + 5x4 = 6 + 5 × 1 = 11,
= 1 − 3x4 + 2x3 = 1 − 3 × 1 + 2 × 11 = 20,
= 8 − 4x4 −