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Modul Bab 3. MATRIKS SMA PAHOA

𝑨.𝑷𝒆𝒏𝒈𝒆𝒓𝒕𝒊𝒂𝒏 Matriks = susunan bilangan dalam baris & kolom yg

berbentuk persegi atau persegi panjang Ukuran suatu matriks ditentukan dengan ORDO Ordo ditulis dengan: 𝑏𝑎𝑛𝑦𝑎𝑘𝑏𝑎𝑟𝑖𝑠 × 𝑏𝑎𝑛𝑦𝑎𝑘𝑘𝑜𝑙𝑜𝑚 Banyaknya elemen matriks ordo 2 × 1 adalah 2 buah Banyaknya elemen matriks ordo 4 × 3 adalah 12 buah Contoh:

1. Matriks 𝐴 = >5 9−8 01 −7

E → 𝑎𝑑𝑎3𝑏𝑎𝑟𝑖𝑠&2𝑘𝑜𝑙𝑜𝑚

→ 𝑑𝑖𝑠𝑒𝑏𝑢𝑡𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑜𝑟𝑑𝑜3 × 2 → 𝑏𝑎𝑟𝑖𝑠𝐼:5&9 → 𝑘𝑜𝑙𝑜𝑚𝐼𝐼:9, 0, −7

2. Matriks 𝐵 = P 1 0 4−3 4 7Q → 2𝑏𝑎𝑟𝑖𝑠&3𝑘𝑜𝑙𝑜𝑚

→ 𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑜𝑟𝑑𝑜2 × 3

3. Jika 𝐶 = >1 −2−3 45 −6

E 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑐UV + 𝑐XU =?

𝑐UV + 𝑐XU = −2 + 5 = 3

4. Jika 𝐸 = >0 9 −34 −2 −85 1 −1

E 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑒XV − 𝑒VX =?

𝑑XV + 𝑑VX = 1 − (−8) = 9 Latihan 1: 1. Tentukan ordo matriks:

𝑎)(36) 𝑏)>

2 −11 15 −3 912 4 −4

E

2. 𝐽𝑖𝑘𝑎𝑃 = P5 6 09 −4 3Q 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛:

𝑎)𝑝VU + 𝑝UX =

𝑏)𝑝UV − 2𝑝VV =

3. 𝐽𝑖𝑘𝑎𝑀 = a4 5 −1−3 8 710

612

−52

b 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛:

𝑎)4𝑚cX − 𝑚VU =

𝑏)𝑚XV − 𝑚VX =

𝑐)𝑚cV + 2𝑚UX =

Nama: kelas: 11

𝑩.𝑻𝒓𝒂𝒏𝒔𝒑𝒐𝒔𝒆𝑻 𝑇𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑒 ≈ 𝑝𝑒𝑟𝑢𝑏𝑎ℎ𝑎𝑛𝑏𝑒𝑛𝑡𝑢𝑘/𝑠𝑢𝑠𝑢𝑛𝑎𝑛

𝑃𝑎𝑑𝑎𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑝𝑒𝑟𝑠𝑒𝑔𝑖(𝑗𝑢𝑚𝑙𝑎ℎ𝑏𝑎𝑟𝑖𝑠 = 𝑗𝑢𝑚𝑘𝑜𝑙𝑜𝑚) 𝑒𝑙𝑒𝑚𝑒𝑛𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙𝑡𝑒𝑡𝑎𝑝; 𝑏𝑒𝑟𝑙𝑎𝑘𝑢𝑠𝑒𝑝𝑒𝑟𝑡𝑖𝑐𝑒𝑟𝑚𝑖𝑛

𝐴 = P𝒂 𝑏𝑐 𝒅Q → 𝐴

𝑻 = P𝒂 𝑐𝑏 𝒅Q

𝐵 = >𝒂 𝑏 𝑐𝑑 𝒆 𝑓𝑔 ℎ 𝒊

E → 𝐵𝑻 = >𝒂 𝑑 𝑔𝑏 𝒆 ℎ𝑐 𝑓 𝒊

E

𝐶 = >𝑎 𝑏𝑐 𝑑𝑒 𝑓

E → 𝐶𝑻 = P𝑎 𝑐 𝑒𝑏 𝑑 𝑓Q

Contoh:

1.𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑒𝐴 = P𝟑 97 𝟐Q

𝐽𝑎𝑤𝑎𝑏:𝐴𝑻 = P𝟑 79 𝟐Q

2.𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑒𝐵 = >𝟐 1 89 𝟎 43 7 𝟔

E

𝐽𝑎𝑤𝑎𝑏:𝐵𝑻 = >𝟐 9 31 𝟎 78 4 𝟔

E

3.𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑒𝐶 = P8 0 26 1 4Q

𝐽𝑎𝑤𝑎𝑏:𝐶𝑻 = >8 60 12 4

E

𝑪.𝑲𝒆𝒔𝒂𝒎𝒂𝒂𝒏 𝐷𝑢𝑎𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑑𝑖𝑘𝑎𝑡𝑎𝑘𝑎𝑛𝑠𝑎𝑚𝑎𝑗𝑖𝑘𝑎&ℎ𝑎𝑛𝑦𝑎𝑗𝑖𝑘𝑎 𝑒𝑙𝑒𝑚𝑒𝑛𝑦𝑔𝑏𝑒𝑟𝑠𝑒𝑠𝑢𝑎𝑖𝑎𝑛𝑝𝑒𝑟𝑠𝑖𝑠𝑠𝑎𝑚𝑎.

Contoh:

1.𝐽𝑖𝑘𝑎 >2 𝑎 − 1

𝑏 + 4 0𝑐 2𝑏

E = >2 8𝑎 0

2𝑏 − 2 𝑑E

𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑎 + 𝑏 + 𝑐 − 𝑑 =?

𝐽𝑎𝑤𝑎𝑏:

𝑎 − 1 = 8 → 𝑎 = 9 𝑐 = 2. 5 − 2 = 8

𝑏 + 4 = 9 → 𝑏 = 5 2. 5 = 𝑑 = 10

𝑎 + 𝑏 + 𝑐 − 𝑑 = 9 + 5 + 8 − 10 = 12

2

2. P12 − 𝑎 82 𝑏VQ = P7 8

2 9Q → 𝑎 + 𝑏 =?

𝐽𝑎𝑤𝑎𝑏:

12 − 𝑎 = 7 → 𝑎 = 5

𝑏V = 9 → 𝑏 = −3 ∨ 𝑏 = 3

𝑚𝑎𝑘𝑎 |𝑎 + 𝑏 = 5 − 3 = 2

𝑎𝑡𝑎𝑢𝑎 + 𝑏 = 5 + 3 = 8

Latihan 2:

1. 𝐽𝑖𝑘𝑎 P2𝑎 16 2𝑏 − 53 −9 𝑐 Q = P20 2𝑐 1

3 −9 𝑑 − 4Q

𝑚𝑎𝑘𝑎𝑎 + 𝑏 + 𝑐 − 𝑑 = _____

2. 𝐽𝑖𝑘𝑎𝐴 = P−4 𝑏 − 12𝑐 11 − 𝑎Q; 𝐵 = P−4 𝑎 + 2

7 5 Q

𝑑𝑎𝑛𝐴 = 𝐵𝑻𝑚𝑎𝑘𝑎𝑎 + 𝑏 + 𝑐 = _____

3. 𝐽𝑖𝑘𝑎 P2 𝑎V − 10𝑏 5

Q = P2 3𝑎8 5 Q

𝑚𝑎𝑘𝑎𝑎 − 𝑏 = ______________

𝑫.𝑶𝒑𝒆𝒓𝒂𝒔𝒊𝑴𝒂𝒕𝒓𝒊𝒌𝒔𝑂𝑝𝑒𝑟𝑎𝑠𝑖𝑎𝑛𝑡𝑎𝑟𝑚𝑎𝑡𝑟𝑖𝑘𝑠ℎ𝑎𝑛𝑦𝑎𝑎𝑑𝑎:+; −;×𝑇𝑖𝑑𝑎𝑘𝑎𝑑𝑎𝑜𝑝𝑒𝑟𝑎𝑠𝑖𝑝𝑒𝑚𝑏𝑎𝑔𝑖𝑎𝑛𝑚𝑎𝑡𝑟𝑖𝑘𝑠

𝑫𝟏.𝑷𝒆𝒏𝒋𝒖𝒎𝒍𝒂𝒉𝒂𝒏&𝑷𝒆𝒏𝒈𝒖𝒓𝒂𝒏𝒈𝒂𝒏𝑆𝑦𝑎𝑟𝑎𝑡𝑛𝑦𝑎:𝑜𝑟𝑑𝑜𝑘𝑒𝑑𝑢𝑎𝑚𝑎𝑡𝑟𝑖𝑘𝑠ℎ𝑎𝑟𝑢𝑠𝑠𝑎𝑚𝑎.𝐸𝑙𝑒𝑚𝑒𝑛𝑦𝑔𝑏𝑒𝑟𝑠𝑒𝑠𝑢𝑎𝑖𝑎𝑛𝑑𝑖𝑡𝑎𝑚𝑏𝑎ℎ/𝑑𝑖𝑘𝑢𝑟𝑎𝑛𝑔𝑖.

Contoh:

1.𝐻𝑖𝑡𝑢𝑛𝑔𝑙𝑎ℎ P 6 8−1 7Q + P

11 −41 2 Q =?

𝐽𝑎𝑤𝑎𝑏: = P17 40 9Q

2.𝐻𝑖𝑡𝑢𝑛𝑔𝑙𝑎ℎ P 6 8−1 7Q − P

11 −41 2 Q =?

𝐽𝑎𝑤𝑎𝑏: = P−5 12−2 5 Q

3.𝐻𝑖𝑡𝑢𝑛𝑔𝑙𝑎ℎ P9 25 1Q − P

34Q =?

𝐽𝑎𝑤𝑎𝑏:𝒕𝒊𝒅𝒂𝒌𝒃𝒊𝒔𝒂

Latihan 3:

1. 𝐻𝑖𝑡𝑢𝑛𝑔𝑙𝑎ℎ >2 −34 −5−6 7

E + >7 −13 4−2 1

E = > E

2. 𝐻𝑖𝑡𝑢𝑛𝑔𝑙𝑎ℎ >2 −34 −5−6 7

E − >7 −13 4−2 1

E = > E

3. 𝑆 = P4 −90 2 Q &𝐻 = P3 −2

8 −5Q 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛

𝑆 + 𝐻𝑻 =

4. 𝐴 = P 5 −1 −210 3 7 Q &𝑅 = >

2 8−1 40 9

E 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛

𝐴𝑻 − 𝑅 =

𝑫𝟐.𝑷𝒆𝒓𝒌𝒂𝒍𝒊𝒂𝒏𝐴𝑑𝑎2𝑚𝑎𝑐𝑎𝑚, 𝑦𝑎𝑖𝑡𝑢:

𝑠𝑘𝑎𝑙𝑎𝑟 × 𝑚𝑎𝑡𝑟𝑖𝑘𝑠&𝑚𝑎𝑡𝑟𝑖𝑘𝑠 × 𝑚𝑎𝑡𝑟𝑖𝑘𝑠

𝒂)𝑺𝒌𝒂𝒍𝒂𝒓 × 𝑴𝒂𝒕𝒓𝒊𝒌𝒔

𝑆𝑒𝑚𝑢𝑎𝑒𝑙𝑒𝑚𝑒𝑛𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑑𝑖𝑘𝑎𝑙𝑖𝑘𝑎𝑛𝑑𝑒𝑛𝑔𝑎𝑛𝑠𝑘𝑎𝑙𝑎𝑟

Contoh:

1.𝐽𝑖𝑘𝑎𝐺 = P8 05 −6Q 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛

12𝐺

𝐽𝑎𝑤𝑎𝑏:12𝐺 =

12 P8 05 −6Q = �

4 05/2 −3�

2.𝐽𝑖𝑘𝑎𝑈 = P8 05 −4Q &𝑉 = P3 −1

6 7 Q

𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑈� − 2𝑉

𝐽𝑎𝑤𝑎𝑏:𝑈𝑻 − 2𝑉 = P8 50 −4Q − 2P

3 −16 7 Q

= P8 50 −4Q − P

6 −212 14Q = P

2 7−12 −18Q

3

𝒃).𝑴𝒂𝒕𝒓𝒊𝒌𝒔 × 𝑴𝒂𝒕𝒓𝒊𝒌𝒔

𝑃𝑒𝑟𝑘𝑎𝑙𝑖𝑎𝑛𝑎𝑛𝑡𝑎𝑟𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑚𝑒𝑛𝑔𝑔𝑢𝑛𝑎𝑘𝑎𝑛𝑝𝑟𝑖𝑛𝑠𝑖𝑝: "𝑩𝑨𝑹𝑰𝑺𝒌𝒂𝒍𝒊𝑲𝑶𝑳𝑶𝑴"

"𝐽𝑢𝑚𝑙𝑎ℎ𝑘𝑜𝑙𝑜𝑚𝑝𝑎𝑑𝑎𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑝𝑒𝑟𝑡𝑎𝑚𝑎 𝐻𝐴𝑅𝑈𝑆𝑠𝑎𝑚𝑎𝑑𝑒𝑛𝑔𝑎𝑛

𝑗𝑢𝑚𝑙𝑎ℎ𝑏𝑎𝑟𝑖𝑠𝑝𝑎𝑑𝑎𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑘𝑒𝑑𝑢𝑎" 𝐽𝑖𝑘𝑎𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑜𝑟𝑑𝑜𝟐 × 𝟐𝑑𝑖𝑘𝑎𝑙𝑖𝑘𝑎𝑛𝑑𝑒𝑛𝑔𝑎𝑛𝑜𝑟𝑑𝑜𝟐 × 𝟐 𝑚𝑎𝑘𝑎ℎ𝑎𝑠𝑖𝑙𝑛𝑦𝑎𝑎𝑑𝑎𝑙𝑎ℎ𝑜𝑟𝑑𝑜𝟐 × 𝟐 ∗ 𝟐 × 𝟐 = 𝟐 × 𝟐

𝐽𝑖𝑘𝑎𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑜𝑟𝑑𝑜𝟐 × 𝟑𝑑𝑖𝑘𝑎𝑙𝑖𝑘𝑎𝑛𝑑𝑒𝑛𝑔𝑎𝑛𝑜𝑟𝑑𝑜𝟑 × 𝟏 𝑚𝑎𝑘𝑎ℎ𝑎𝑠𝑖𝑙𝑛𝑦𝑎𝑎𝑑𝑎𝑙𝑎ℎ𝑜𝑟𝑑𝑜𝟐 × 𝟑 ∗ 𝟑 × 𝟏 = 𝟐 × 𝟏

𝐽𝑖𝑘𝑎𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑜𝑟𝑑𝑜𝟒 × 𝟑𝑑𝑖𝑘𝑎𝑙𝑖𝑘𝑎𝑛𝑑𝑒𝑛𝑔𝑎𝑛𝑜𝑟𝑑𝑜𝟐 × 𝟏 𝑚𝑎𝑘𝑎𝒕𝒊𝒅𝒂𝒌𝒃𝒊𝒔𝒂𝒅𝒊𝒉𝒊𝒕𝒖𝒏𝒈,

𝑘𝑎𝑟𝑒𝑛𝑎𝑗𝑢𝑚𝑙𝑎ℎ𝑘𝑜𝑙𝑜𝑚𝑝𝑎𝑑𝑎𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑝𝑒𝑟𝑡𝑎𝑚𝑎(= 𝟑) ≠ 𝑗𝑢𝑚𝑙𝑎ℎ𝑏𝑎𝑟𝑖𝑠𝑝𝑎𝑑𝑎𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑘𝑒𝑑𝑢𝑎(= 𝟐)

Contoh:

1.𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛 P𝑎 𝑏𝑐 𝑑Q𝟐×𝟐

× �𝑒 𝑓𝑔 ℎ�𝟐×𝟐

𝐽𝑎𝑤𝑎𝑏: = �𝑎𝑒 + 𝑏𝑔 𝑎𝑓 + 𝑏ℎ𝑐𝑒 + 𝑑𝑔 𝑐𝑓 + 𝑑ℎ�𝟐×𝟐

2.𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛 P4 −21 3 Q × P

1 95 6Q

𝐽𝑎𝑤𝑎𝑏: = P4 ∗ 1 +−2 ∗ 5 4 ∗ 9 +−2 ∗ 61 ∗ 1 + 3 ∗ 5 1 ∗ 9 + 3 ∗ 6Q

= P−6 2416 27Q𝟐×𝟐

3.𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛 P7 3

5 −4Q × P0 −21 8 Q

𝐽𝑎𝑤𝑎𝑏: = P7 ∗ 0 + 3 ∗ 1 7 ∗ −2 + 3 ∗ 85 ∗ 0 +−4 ∗ 1 5 ∗ −2 +−4 ∗ 8Q

= P 3 10−4 −42Q𝟐×𝟐

4.𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛 >3 0−4 51 2

E𝟑×𝟐

× P 2 7−1 6Q𝟐×𝟐

𝐽𝑎𝑤𝑎𝑏: = >6 − 0 21 + 0−8 − 5 −28 + 302 − 2 7 + 12

E𝟑×𝟐

= >6 21−13 20 19

E𝟑×𝟐

5.𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛 P−7 5 12 4 −2Q𝟐×𝟑

× >364E𝟑×𝟏

𝐽𝑎𝑤𝑎𝑏: = P−21 + 30 + 46 + 24 − 8 Q𝟐×𝟏

= P1322Q𝟐×𝟏

6.𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛 P−7 5 12 4 −2Q𝟐×𝟑

× P3 20 1Q𝟐×𝟐

𝐽𝑎𝑤𝑎𝑏: 𝒕𝒊𝒅𝒂𝒌𝒃𝒊𝒔𝒂

Latihan 3:

1. P3 14 2Q×P

6 −21 5 Q = > E

2. P6 −21 5 Q ×P

3 14 2Q = > E

3. >6 −1−4 20 1

E×P2 14 5Q = � �

4. >6 −1−4 20 1

E×P27Q = > E

5. (4 1 5) ×>−1 32 −61 7

E = P Q

6. P 8 2 0−1 3 4Q ×>

−1 32 −61 7

E = P Q

7. >4 1−1 90 2

E×P3 1 45 0 2Q = � �

8. >2 1 40 3 25 2 1

E×>3 1 02 3 24 0 1

E = � �

9. >2 1 −2−1 3 35 2 1

E>−3 1 02 3 −1−1 0 1

E = � �

4

𝑬.𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒏&𝑰𝒏𝒗𝒆𝒓𝒔𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝐴𝑑𝑖𝑡𝑢𝑙𝑖𝑠:|𝐴|𝑎𝑡𝑎𝑢 𝑑𝑒𝑡 𝐴𝑀𝑎𝑡𝑟𝑖𝑘𝑠𝑆𝑖𝑛𝑔𝑢𝑙𝑎𝑟 → 𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑦𝑔𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝑛𝑦𝑎0

𝐾𝑎𝑟𝑒𝑛𝑎𝑡𝑖𝑑𝑎𝑘𝑎𝑑𝑎𝑜𝑝𝑒𝑟𝑎𝑠𝑖𝑃𝑒𝑚𝑏𝑎𝑔𝑖𝑎𝑛𝑀𝑎𝑡𝑟𝑖𝑘𝑠,𝑚𝑎𝑘𝑎𝑢𝑛𝑡𝑢𝑘𝑚𝑒𝑛𝑐𝑎𝑟𝑖𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑿𝑝𝑎𝑑𝑎𝑝𝑒𝑟𝑠𝑎𝑚𝑎𝑎𝑛𝐴. 𝑿 = 𝐵𝑑𝑖𝑔𝑢𝑛𝑎𝑘𝑎𝑛𝑐𝑎𝑟𝑎𝐼𝑛𝑣𝑒𝑟𝑠𝑀𝑎𝑡𝑟𝑖𝑘𝑠.

𝑬𝟏.𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒏&𝑰𝒏𝒗𝒆𝒓𝒔𝑴𝒂𝒕𝒓𝒊𝒌𝒔𝟐 × 𝟐

𝐽𝑖𝑘𝑎𝐴 = P𝑎 𝑏𝑐 𝑑Q|

𝑑𝑒𝑡 𝐴 = 𝑎𝑑 − 𝑏𝑐

𝐴¢U =1

det 𝐴 P𝒅 −𝒃−𝒄 𝒂 Q

(𝒂𝒅𝑡𝑢𝑘𝑎𝑟𝑝𝑜𝑠𝑖𝑠𝑖; 𝒃𝒄𝑘𝑎𝑙𝑖𝑚𝑖𝑛𝑢𝑠)

Contoh:

1. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝐴 = P4 26 7Q

𝐽𝑎𝑤𝑎𝑏:|𝐴| = 4 ∗ 7 − 2 ∗ 6 = 28 − 12 = 16

2. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝐵 = P4 −82 −3Q

𝐽𝑎𝑤𝑎𝑏: 𝑑𝑒𝑡 𝐵 = 4 ∗ −3 −(−8 ∗ 2) = 4

3. 𝐶 = P𝑎 − 3 94 2Q &𝐶𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟 → 𝑎 =?

𝐽𝑎𝑤𝑎𝑏:𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟 → 𝑑𝑒𝑡 𝐶 = 0

(𝑎 − 3) ∗ 2 − 9 ∗ 4 = 0

2(𝑎 − 3) = 36 → 𝑎 = 21

4. 𝐷 = P𝑚 − 3 25 𝑚Q &𝐷𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟 → 𝑚 =?

𝐽𝑎𝑤𝑎𝑏:

(𝑚 − 3) ∗ 𝑚 − 2 ∗ 5 = 0

𝑚V − 3𝑚 − 10 = 0 → (𝑚 + 2)(𝑚 − 5) = 0

𝑚 = −2 ∨ 𝑚 = 5

5. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑖𝑛𝑣𝑒𝑟𝑠𝐸 = P7 35 2Q

𝐽𝑎𝑤𝑎𝑏:

|𝐸| = 14 − 15 = −1

𝐸¢U =1−1

P 𝟐 −𝟑−𝟓 𝟕 Q = P−2 3

5 −7Q

𝒄𝒆𝒌:𝑘𝑎𝑙𝑖𝑘𝑎𝑛𝑠𝑎𝑗𝑎𝐸𝑑𝑒𝑛𝑔𝑎𝑛𝐸¢U𝑎𝑡𝑎𝑢𝑠𝑒𝑏𝑎𝑙𝑖𝑘𝑛𝑦𝑎

𝑝𝑎𝑑𝑎𝑠𝑜𝑎𝑙𝑡𝑎𝑑𝑖:𝐸 = P7 35 2Q &𝐸

¢U = P−2 35 −7Q

𝑐𝑜𝑏𝑎𝑘𝑖𝑡𝑎𝑘𝑎𝑙𝑖𝑘𝑎𝑛:

𝐸¢U × 𝐸 = P−2 35 −7Q × P

7 35 2Q

= P−14 + 15 −6 + 635 − 35 15 − 14Q = P1 0

0 1Q 𝑜𝑘𝑒

𝑀𝑎𝑡𝑟𝑖𝑘𝑠 P1 00 1Q 𝑑𝑖𝑠𝑒𝑏𝑢𝑡𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝐼𝑛𝑑𝑒𝑛𝑡𝑖𝑡𝑎𝑠

(𝑰)

𝑀𝑎𝑡𝑟𝑖𝑘𝑠𝑦𝑔𝑑𝑖𝑘𝑎𝑙𝑖𝑘𝑎𝑛𝑑𝑔𝑛𝑰𝑡𝑖𝑑𝑎𝑘𝑎𝑘𝑎𝑛𝑏𝑒𝑟𝑢𝑏𝑎ℎ,

𝑚𝑖𝑠𝑎𝑙𝑛𝑦𝑎 P9 32 6Q × P

1 00 1Q = P9 3

2 6Q

6. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑖𝑛𝑣𝑒𝑟𝑠𝐹 = P 4 5−6 −8Q

𝐽𝑎𝑤𝑎𝑏:

𝑑𝑒𝑡 𝐹 = −32 − (−30) = −2

𝐹¢U =1−2

P−𝟖 −𝟓𝟔 𝟒 Q = P 4 5/2

−3 −2Q

𝒄𝒆𝒌:𝐹 × 𝐹¢U = P 4 5−6 −8Q × P

4 5/2−3 −2Q

= P 16 − 15 10 − 10−24 + 24 −15 + 16Q = P1 0

0 1Q 𝑜𝑘𝑒

7. 𝐽𝑖𝑘𝑎𝐺 = P−9 5−7 4Q &𝐻 = P−4 5

−7 9Q 𝑏𝑢𝑘𝑡𝑖𝑘𝑎𝑛

𝑏𝑎ℎ𝑤𝑎𝐺&𝐻𝑎𝑑𝑎𝑙𝑎ℎ2𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑦𝑔𝑠𝑎𝑙𝑖𝑛𝑔𝑖𝑛𝑣𝑒𝑟𝑠

𝐽𝑎𝑤𝑎𝑏:

𝑐𝑒𝑘𝑠𝑎𝑗𝑎𝑖𝑛𝑣𝑒𝑟𝑠𝑑𝑎𝑟𝑖𝑠𝑎𝑙𝑎ℎ𝑠𝑎𝑡𝑢𝑚𝑎𝑡𝑟𝑖𝑘𝑠

|𝐺| = −36 − (−35) = −1

𝐺¢U =1−1

P𝟒 −𝟓𝟕 −𝟗Q = P−4 5

−7 9Q = 𝐻𝑡𝑒𝑟𝑏𝑢𝑘𝑡𝑖

𝑐𝑎𝑟𝑎𝑙𝑎𝑖𝑛:𝑘𝑎𝑙𝑖𝑘𝑎𝑛𝑘𝑒𝑑𝑢𝑎𝑚𝑎𝑡𝑟𝑖𝑘𝑠

P−9 5−7 4Q × P

−4 5−7 9Q = P36 − 35 −45 + 45

28 − 28 −35 + 36Q

= P1 00 1Q 𝑜𝑘𝑒

8. 𝐽𝑖𝑘𝑎𝑊 = P2 30 −1Q 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛

(𝑊V)¢U

𝐽𝑎𝑤𝑎𝑏:

𝑊V = 𝑊 ∗𝑊 = P2 30 −1Q P

2 30 −1Q = P4 3

0 1Q

𝑑𝑒𝑡(𝑊V) = 4 − 0 = 4

𝑖𝑛𝑣𝑒𝑟𝑠:(𝑊V)¢U =14P𝟏 −𝟑𝟎 𝟒 Q = P1/4 −3/4

0 1 Q

5

Latihan 4:

1. 𝐻𝑖𝑡𝑢𝑛𝑔𝑙𝑎ℎ ­4 93 5­ =

2. 𝐽𝑖𝑘𝑎𝑀 = P3 42 5Q 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛|𝑀|

3. 𝐽𝑖𝑘𝑎𝑀 = P3 42 5Q 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛®𝑀

𝑻®

𝑀� =→ ®𝑀𝑻® =

4. 𝐽𝑖𝑘𝑎 P3 𝑎 + 52 8 Q 𝑎𝑑𝑎𝑙𝑎ℎ𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟 → 𝑎 =?

5. 𝐽𝑖𝑘𝑎 P4 𝑎 + 5𝑎 9 Q 𝑎𝑑𝑎𝑙𝑎ℎ𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟 → 𝑎 =?

6. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑖𝑛𝑣𝑒𝑟𝑠𝐶 = P 5 7−3 −4Q

7. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑖𝑛𝑣𝑒𝑟𝑠𝐷 = P−1 4−2 6Q

8. 𝐽𝑖𝑘𝑎𝐸 = P−3 2−7 5Q &𝐹 = P−5 2

−7 3Q 𝑏𝑢𝑘𝑡𝑖𝑘𝑎𝑛

𝑏𝑎ℎ𝑤𝑎𝐸&𝐹𝑠𝑎𝑙𝑖𝑛𝑔𝑖𝑛𝑣𝑒𝑟𝑠

9. 𝐽𝑖𝑘𝑎𝐹 = P3 44 5Q &𝐺 = P2 3

1 1Q 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛:

𝑎)𝐹𝐺 = P QP Q = P Q

𝑏)𝐺¢U. 𝐹¢U

𝑹𝒖𝒎𝒖𝒔𝑰𝒏𝒗𝒆𝒓𝒔(𝑢𝑛𝑡𝑢𝑘𝑚𝑒𝑛𝑐𝑎𝑟𝑖𝑿)

𝑗𝑎𝑛𝑔𝑎𝑛𝑡𝑒𝑟𝑏𝑎𝑙𝑖𝑘 ∶)

Contoh:

1. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑿𝑗𝑖𝑘𝑎 P1 −12 1 Q .𝑿 = P5 −2

7 11Q

𝐽𝑎𝑤𝑎𝑏:

𝐶𝑎𝑟𝑎1: 𝑃𝑒𝑚𝑖𝑠𝑎𝑙𝑎𝑛

P1 −12 1 Q. P

𝑎 𝑏𝑐 𝑑Q = P5 −2

7 11Q

𝑎 − 𝑐 = 52𝑎 + 𝑐 = 7° 𝑎 = 4; 𝑐 = −1

𝑏 − 𝑑 = −22𝑏 + 𝑑 = 11° 𝑏 = 3; 𝑑 = 5

𝑚𝑎𝑘𝑎𝑿 = P𝑎 𝑏𝑐 𝑑Q = P 4 3

−1 5Q

𝐶𝑎𝑟𝑎2: 𝑅𝑢𝑚𝑢𝑠𝐼𝑛𝑣𝑒𝑟𝑠

P1 −12 1 Q .𝑿 = P5 −2

7 11Q

𝐴.𝑿 = 𝐵 → 𝑿 = 𝐴¢U. 𝐵

P1 −12 1 Q → 𝑑𝑒𝑡 = 1 − (−2) = 3

𝑖𝑛𝑣𝑒𝑟𝑠:13P 𝟏 𝟏−𝟐 𝟏Q 𝑖𝑛𝑔𝑎𝑡:𝑡𝑢𝑘𝑎𝑟𝑝𝑜𝑠𝑖𝑠𝑖; 𝑘𝑎𝑙𝑖𝑚𝑖𝑛𝑢𝑠

𝑿 = 𝐴¢U. 𝐵 = 13P 𝟏 𝟏−𝟐 𝟏Q. P

5 −27 11Q

= 13P 5 + 7 −2 + 11−10 + 7 4 + 11 Q

= 13P12 9−3 15Q = P

4 3−1 5Q

2. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑿𝑗𝑖𝑘𝑎 P 4 −3−7 5 Q .𝑿 = P 22 23

−37 −41Q

𝐽𝑎𝑤𝑎𝑏:

𝐶𝑎𝑟𝑎1: 𝑃𝑒𝑚𝑖𝑠𝑎𝑙𝑎𝑛

P 4 −3−7 5 Q. P

𝑎 𝑏𝑐 𝑑Q = P 22 23

−37 −41Q

4𝑎 − 3𝑐 = 22−7𝑎 + 5𝑐 = −37° 𝑎 = 1; 𝑐 = −6

4𝑏 − 3𝑑 = 23−7𝑏 + 5𝑑 = −41° 𝑏 = 8; 𝑑 = 3

𝑚𝑎𝑘𝑎𝑿 = P𝑎 𝑏𝑐 𝑑Q = P 1 8

−6 3Q

𝐴. 𝑿 = 𝐵 → 𝑿 = 𝐴¢U. 𝐵

𝑿. 𝐴 = 𝐵 → 𝑿 = 𝐵. 𝐴¢U

6

𝐶𝑎𝑟𝑎2: 𝑅𝑢𝑚𝑢𝑠𝐼𝑛𝑣𝑒𝑟𝑠

P 4 −3−7 5 Q .𝑿 = P 22 23

−37 −41Q

𝐴.𝑿 = 𝐵 → 𝑿 = 𝐴¢U. 𝐵

P 4 −3−7 5 Q → 𝑑𝑒𝑡 = 20 − 21 = −1

𝑖𝑛𝑣𝑒𝑟𝑠:1−1

P𝟓 𝟑𝟕 𝟒Q 𝑖𝑛𝑔𝑎𝑡:𝑡𝑢𝑘𝑎𝑟𝑝𝑜𝑠𝑖𝑠𝑖; 𝑘𝑎𝑙𝑖𝑚𝑖𝑛𝑢𝑠

𝑿 = 𝐴¢U. 𝐵 = 1−1

P𝟓 𝟑𝟕 𝟒Q. P

22 23−37 −41Q

= 1−1

P110 − 111 115 − 123154 − 148 161 − 164Q

= 1−1

P−1 −86 −3Q = P

1 8−6 3Q

3. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑿𝑗𝑖𝑘𝑎𝑿. P4 21 −1Q = P17 13

9 3 Q

𝐽𝑎𝑤𝑎𝑏:

𝐶𝑎𝑟𝑎1: 𝑃𝑒𝑚𝑖𝑠𝑎𝑙𝑎𝑛

P𝑎 𝑏𝑐 𝑑Q P

4 21 −1Q = P17 13

9 3 Q

4𝑎 + 𝑏 = 172𝑎 − 𝑏 = 13° 𝑎 = 5; 𝑏 = −3

4𝑐 + 𝑑 = 92𝑐 − 𝑑 = 3° 𝑐 = 2; 𝑑 = 1

𝑚𝑎𝑘𝑎𝑿 = P𝑎 𝑏𝑐 𝑑Q = P5 −3

2 1 Q

𝐶𝑎𝑟𝑎2: 𝑅𝑢𝑚𝑢𝑠𝐼𝑛𝑣𝑒𝑟𝑠

𝑿. P4 21 −1Q = P17 13

9 3 Q

𝑿.𝐴 = 𝐵 → 𝑿 = 𝐵. 𝐴¢U

P4 21 −1Q → 𝑑𝑒𝑡 = −4 − 2 = −6

𝑖𝑛𝑣𝑒𝑟𝑠:1−6

P−𝟏 −𝟐−𝟏 𝟒 Q 𝑖𝑛𝑔𝑎𝑡: 𝑡𝑢𝑘𝑎𝑟𝑝𝑜𝑠𝑖𝑠𝑖; 𝑘𝑎𝑙𝑖𝑚𝑖𝑛𝑢𝑠

𝑿 = 𝐵. 𝐴¢U = P17 139 3 Q.

1−6

P−𝟏 −𝟐−𝟏 𝟒 Q

= 1−6

P−17 − 13 −34 + 52−9 − 3 −18 + 12Q

= 1−6

P−30 18−12 −6Q = P

5 −32 1 Q

4. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑿𝑗𝑖𝑘𝑎𝑿. P 5 −711 8 Q = P27 9

59 11Q

𝐽𝑎𝑤𝑎𝑏:

𝐶𝑎𝑟𝑎1: 𝑃𝑒𝑚𝑖𝑠𝑎𝑙𝑎𝑛

P𝑎 𝑏𝑐 𝑑Q P

5 −711 8 Q = P27 9

59 11Q

5𝑎 + 11𝑏 = 27−7𝑎 + 8𝑏 = 9° 𝑎 = 1; 𝑏 = 2

5𝑐 + 11𝑑 = 59−7𝑐 + 8𝑑 = 11° 𝑐 = 3; 𝑑 = 4

𝑚𝑎𝑘𝑎𝑿 = P𝑎 𝑏𝑐 𝑑Q = P1 2

3 4Q

𝐶𝑎𝑟𝑎2: 𝑅𝑢𝑚𝑢𝑠𝐼𝑛𝑣𝑒𝑟𝑠

𝑿. P 5 −711 8 Q = P27 9

59 11Q

𝑿.𝐴 = 𝐵 → 𝑿 = 𝐵. 𝐴¢U

P 5 −711 8 Q → 𝑑𝑒𝑡 = 40 − (−77) = 117

𝑖𝑛𝑣𝑒𝑟𝑠:1117

P 𝟖 𝟕−𝟏𝟏 𝟓Q 𝑡𝑢𝑘𝑎𝑟𝑝𝑜𝑠𝑖𝑠𝑖; 𝑘𝑎𝑙𝑖𝑚𝑖𝑛𝑢𝑠

𝑿 = 𝐵. 𝐴¢U = P27 959 11Q.

1117

P 𝟖 𝟕−𝟏𝟏 𝟓Q

= 1117

P 216 − 99 189 + 45472 − 121 413 + 55Q

= 1117

P117 234351 468Q = P

1 23 4Q

Latihan 5:

1. 𝐷𝑒𝑛𝑔𝑎𝑛𝑐𝑎𝑟𝑎𝑝𝑒𝑚𝑖𝑠𝑎𝑙𝑎𝑛, 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑿𝑗𝑖𝑘𝑎:

𝑿. P1 21 −1Q = P

7 211 4Q

7

2. 𝐷𝑒𝑛𝑔𝑎𝑛𝑐𝑎𝑟𝑎𝐼𝑛𝑣𝑒𝑟𝑠𝑀𝑎𝑡𝑟𝑖𝑘𝑠, 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑿𝑗𝑖𝑘𝑎:

𝑿. P1 21 −1Q = P

7 211 4Q

3. 𝐷𝑒𝑛𝑔𝑎𝑛𝑐𝑎𝑟𝑎𝑝𝑒𝑚𝑖𝑠𝑎𝑙𝑎𝑛, 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑿𝑗𝑖𝑘𝑎:

P−2 12 1Q .𝑿 = P

−3 −417 20Q

4. 𝐷𝑒𝑛𝑔𝑎𝑛𝑐𝑎𝑟𝑎𝐼𝑛𝑣𝑒𝑟𝑠𝑀𝑎𝑡𝑟𝑖𝑘𝑠, 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑿𝑗𝑖𝑘𝑎:

P−2 12 1Q .𝑿 = P

−3 −417 20Q

5. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑿𝑗𝑖𝑘𝑎𝑿. P−13 1115 −12Q = P

27 −1831 −20Q

𝑬𝟐.𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒏&𝑰𝒏𝒗𝒆𝒓𝒔𝑴𝒂𝒕𝒓𝒊𝒌𝒔𝟑 × 𝟑

𝐽𝑖𝑘𝑎𝐴 = >𝑎 𝑏 𝑐𝑑 𝑒 𝑓𝑔 ℎ 𝑖

E

𝑑𝑒𝑡 𝐴 = 𝑎𝑒𝑖 +𝑏𝑓𝑔 + 𝑐𝑑ℎ − (𝑐𝑒𝑔 + 𝑎𝑓ℎ + 𝑏𝑑𝑖)

Contoh:

1. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝐴 = >9 6 23 1 07 5 4

E

𝐽𝑎𝑤𝑎𝑏:

|𝐴| = 9 ∗ 1 ∗ 4 + 6 ∗ 0 ∗ 7 + 2 ∗ 3 ∗ 5−(2 ∗ 1 ∗ 7 + 9 ∗ 0 ∗ 5 + 6 ∗ 3 ∗ 4)

= 36 + 0 + 30 − (14 + 0 + 72)= 66 − 86 = −20

2. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝐵 = >1 6 23 10 11 5 −2

E

𝐽𝑎𝑤𝑎𝑏:

|𝐵| = −20 + 6 + 30 − (20 + 5 − 36)= 16 − (−11) = 27

3. 𝐽𝑖𝑘𝑎 ±3 1 01 −1 1𝑎 2 −2

± = 6 → 𝑎 =?

𝐽𝑎𝑤𝑎𝑏:

6 + 𝑎 + 0 − (0 + 6 − 2) = 66 + 𝑎 − 4 = 6 → 𝑎 = 4

4. 𝐽𝑖𝑘𝑎 ±5 𝑚 32 −1 16 7 4

± = 9 → 𝑚 =?

𝐽𝑎𝑤𝑎𝑏:

−20 + 6𝑚 + 42 − (−18 + 35 + 8𝑚) = 96𝑚 + 22 − (8𝑚 + 17) = 9

−2𝑚 + 5 = 9→ 𝑚 = −2

5. 𝐽𝑖𝑘𝑎 ±5 −1 61 0 34 1 7

± = ­4 𝑐2 5­ → 𝑐 =?

𝐽𝑎𝑤𝑎𝑏:

0 − 12 + 6 − (0 + 15 − 7) = 20 − 2𝑐−6 − 8 = 20 − 2𝑐

2𝑐 = 34 → 𝑐 = 17

8

𝑴𝒆𝒏𝒆𝒏𝒕𝒖𝒌𝒂𝒏𝒊𝒏𝒗𝒆𝒓𝒔𝒎𝒂𝒕𝒓𝒊𝒌𝒔𝟑 × 𝟑

𝑠𝑡𝑒𝑝𝑠:(𝒅𝒆𝑡𝒂𝑘𝑖𝑛𝑣𝑒𝑟𝑠) 𝑎)𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛 𝑏)𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑒 𝑐)𝑎𝑑𝑗𝑜𝑖𝑛𝑡 𝑑)𝑘𝑜𝑓𝑎𝑘𝑡𝑜𝑟 𝑒)𝑖𝑛𝑣𝑒𝑟𝑠

Contoh:

1. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑖𝑛𝑣𝑒𝑟𝑠𝐴 = >1 2 30 −1 23 7 5

E

𝐽𝑎𝑤𝑎𝑏:

𝑎)𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛 𝑑𝑒𝑡 𝐴 = −5 + 12 + 0 − (−9 + 14 + 0) = 2

𝑏)𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑒

𝐴𝑻 = >𝟏 0 32 −𝟏 73 2 𝟓

E

𝑐)𝑎𝑑𝑗𝑜𝑖𝑛𝑡

𝑎𝑑𝑗 =

⎝

⎜⎜⎛

­−1 72 5­ ­2 7

3 5­ ­2 −13 2 ­

­0 32 5­ ­1 3

3 5­ ­1 03 2­

­ 0 3−1 7­ ­1 3

2 7­ ­1 02 −1­

⎠

⎟⎟⎞

= >−19 −11 7−6 −4 23 2 −1

E

𝑑)𝑘𝑜𝑓𝑎𝑘𝑡𝑜𝑟

𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑘𝑜𝑓𝑎𝑘𝑡𝑜𝑟: >+ − +− + −+ − +

E

= >−19 𝟏𝟏 7𝟔 −4 −𝟐3 −𝟏 −1

E

𝑒)𝑖𝑛𝑣𝑒𝑟𝑠

𝑚𝑎𝑘𝑎𝐴¢U = 12>

−19 𝟏𝟏 7𝟔 −4 −𝟐3 −𝟏 −1

E

= >−19/2 11/2 7/23 −2 −13/2 −1/2 −1/2

E

𝒄𝒆𝒌: 𝐴¢U × 𝐴 =12>

−19 𝟏𝟏 7𝟔 −4 −𝟐3 −𝟏 −1

E × >1 2 30 −1 23 7 5

E

=12>

−19 + 0 + 21 −38 − 11 + 49 −57 + 22 + 356 − 0 − 6 12 + 4 − 14 18 − 8 − 103 − 0 − 3 6 + 1 − 7 9 − 2 − 5

E

=12>

2 0 00 2 00 0 2

E = >1 0 00 1 00 0 1

E = 𝑰𝑜𝑘𝑒

2. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑖𝑛𝑣𝑒𝑟𝑠𝐵 = >3 −1 10 4 3−6 5 0

E

𝐽𝑎𝑤𝑎𝑏:

𝑎)𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛 𝑑𝑒𝑡 𝐵 = 0 + 18 + 0 − (−24 + 45 − 0) = −3

𝑏)𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑒

𝐴𝑻 = >𝟑 0 −6−1 𝟒 51 3 𝟎

E

𝑐)𝑎𝑑𝑗𝑜𝑖𝑛𝑡

𝑎𝑑𝑗 =

⎝

⎜⎜⎛

­4 53 0­ ­−1 5

1 0­ ­−1 41 3­

­0 −63 0 ­ ­3 −6

1 0 ­ ­3 01 3­

­0 −64 5 ­ ­ 3 −6

−1 5 ­ ­ 3 0−1 4­

⎠

⎟⎟⎞

= >−15 −5 −718 6 9−24 9 12

E

𝑑)𝑘𝑜𝑓𝑎𝑘𝑡𝑜𝑟

𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑘𝑜𝑓𝑎𝑘𝑡𝑜𝑟: >+ − +− + −+ − +

E

= >−15 𝟓 −7−𝟏𝟖 6 −𝟗24 −𝟗 12

E

𝑒)𝑖𝑛𝑣𝑒𝑟𝑠

𝑚𝑎𝑘𝑎𝐵¢U = 1−3>

−15 𝟓 −7−𝟏𝟖 6 −𝟗24 −𝟗 12

E

= >5 −5/3 7/36 −2 38 3 −4

E

𝒄𝒆𝒌: 𝐵¢U × 𝐵 =1−3>

−15 𝟓 −7−𝟏𝟖 6 −𝟗24 −𝟗 12

E × >3 −1 10 4 3−6 5 0

E

=1−3>

−45 + 42 15 + 20 − 35 −15 + 15−54 + 54 18 + 24 − 45 −18 + 1872 − 72 −24 − 36 + 60 24 − 27

E

=1−3>

−3 0 00 −3 00 0 −3

E = >1 0 00 1 00 0 1

E = 𝑰𝑜𝑘𝑒

9

Latihan 6:

1. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑖𝑛𝑣𝑒𝑟𝑠𝐶 = >0 1 21 −1 03 0 5

E

2. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑖𝑛𝑣𝑒𝑟𝑠𝐷 = >4 3 11 1 −14 3 −1

E

3. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑖𝑛𝑣𝑒𝑟𝑠𝐸 = >5 4 32 1 1−1 6 2

E

10

𝑭.𝑨𝒑𝒍𝒊𝒌𝒂𝒔𝒊𝑴𝒂𝒕𝒓𝒊𝒌𝒔𝐴𝑝𝑙𝑖𝑘𝑎𝑠𝑖𝑚𝑎𝑡𝑟𝑖𝑘𝑠𝑏𝑖𝑎𝑠𝑎𝑛𝑦𝑎𝑑𝑖𝑔𝑢𝑛𝑎𝑘𝑎𝑛𝑢𝑛𝑡𝑢𝑘𝑚𝑒𝑛𝑦𝑒𝑙𝑒𝑠𝑎𝑖𝑘𝑎𝑛𝑠𝑖𝑠𝑡𝑒𝑚𝑝𝑒𝑟𝑠𝑎𝑚𝑎𝑎𝑛𝑙𝑖𝑛𝑒𝑎𝑟&𝑚𝑒𝑛𝑔ℎ𝑖𝑡𝑢𝑛𝑔𝑙𝑢𝑎𝑠𝑑𝑎𝑒𝑟𝑎ℎ𝑝𝑎𝑑𝑎𝑠𝑖𝑠𝑡𝑒𝑚𝑘𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡.

𝑭𝟏.𝑺𝒊𝒔𝒕𝒆𝒎𝑷𝒆𝒓𝒔𝒂𝒎𝒂𝒂𝒏𝑳𝒊𝒏𝒆𝒂𝒓

Contoh:

1. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑝𝑒𝑛𝑦𝑒𝑙𝑒𝑠𝑎𝑖𝑎𝑛𝑑𝑎𝑟𝑖𝑠𝑖𝑠𝑡𝑒𝑚𝑝𝑒𝑟𝑠:

¹3𝑎 − 𝑏 = 112𝑎 + 𝑏 = 9

𝐽𝑎𝑤𝑎𝑏:

𝑎)𝑐𝑎𝑟𝑎𝑖𝑛𝑣𝑒𝑟𝑠:

P3 −12 1 Q P

𝑎𝑏Q = P119 Q

↓ 𝑑𝑒𝑡 = 3 − (−2) = 5

𝑖𝑛𝑣𝑒𝑟𝑠 =15P 𝟏 𝟏−𝟐 𝟑Q

P𝑎𝑏Q =15P 𝟏 𝟏−𝟐 𝟑Q P

119 Q

=15P

11 + 9−22 + 27Q

=15P205 Q = P41Q

𝑑𝑖𝑑𝑎𝑝𝑎𝑡𝑎 = 4&𝑏 = 1

𝑏)𝑐𝑎𝑟𝑎𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛:

P3 −12 1 Q P

𝑎𝑏Q = P𝟏𝟏𝟗 Q

↓ 𝑑𝑒𝑡 = 3 − (−2) = 5

𝑎 =­𝟏𝟏 −1𝟗 1 ­

5=11 − (−9)

5= 4

𝑏 =­3 𝟏𝟏2 𝟗 ­

5=27 − 22

5= 1

2. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑝𝑒𝑛𝑦𝑒𝑙𝑒𝑠𝑎𝑖𝑎𝑛𝑑𝑎𝑟𝑖𝑠𝑖𝑠𝑡𝑒𝑚𝑝𝑒𝑟𝑠:

» 3𝑥 + 4𝑦 = 2−7𝑥 − 6𝑦 = 12

𝐽𝑎𝑤𝑎𝑏:

𝑎)𝑐𝑎𝑟𝑎𝑖𝑛𝑣𝑒𝑟𝑠:

P 3 4−7 −6Q P

𝑥𝑦Q = P 212Q

↓ 𝑑𝑒𝑡 = −18 − (−28) = 10

𝑖𝑛𝑣𝑒𝑟𝑠 =110P−𝟔 −𝟒𝟕 𝟑 Q

P𝑥𝑦Q =

110P−𝟔 −𝟒𝟕 𝟑 Q P

212Q

=110P

−12 − 4814 + 36 Q

=110P

−6050 Q = P−65 Q

𝑑𝑖𝑑𝑎𝑝𝑎𝑡𝑥 = −6&𝑦 = 5

𝑏)𝑐𝑎𝑟𝑎𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛:

P 3 4−7 −6Q P

𝑥𝑦Q = P 𝟐𝟏𝟐Q

↓ 𝑑𝑒𝑡 = −18 − (−28) = 10

𝑥 =­ 𝟐 4𝟏𝟐 −6­

10=−12 − 48

10= −6

𝑦 =­ 3 𝟐−7 𝟏𝟐­

10=36 − (−14)

10= 5

11

𝑭𝟐.𝑳𝒖𝒂𝒔𝑫𝒂𝒆𝒓𝒂𝒉𝐷𝑎𝑝𝑎𝑡𝑚𝑒𝑛𝑔ℎ𝑖𝑡𝑢𝑛𝑔𝑙𝑢𝑎𝑠𝑑𝑎𝑒𝑟𝑎ℎ𝑡𝑒𝑟𝑡𝑢𝑡𝑢𝑝𝑝𝑎𝑑𝑎𝑘𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑘𝑎𝑟𝑡𝑒𝑠𝑖𝑢𝑠(𝑠𝑒𝑔𝑖𝑏𝑒𝑟𝑎𝑝𝑎𝑝𝑢𝑛).

Contoh:

1. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑙𝑢𝑎𝑠𝑑𝑎𝑒𝑟𝑎ℎ𝑦𝑔𝑑𝑖𝑏𝑎𝑡𝑎𝑠𝑖𝑜𝑙𝑒ℎ𝑡𝑖𝑡𝑖𝑘 𝐴(2, 1); 𝐵(9, 1); 𝐶(9, 5)

𝐽𝑎𝑤𝑎𝑏:

𝑐𝑎𝑟𝑎𝑚𝑎𝑛𝑢𝑎𝑙:

𝐿𝑢𝑎𝑠 =12. 7. 4 = 14𝑠𝑎𝑡𝑢𝑎𝑛

𝑝𝑎𝑘𝑎𝑖𝑚𝑎𝑡𝑟𝑖𝑘𝑠:

𝐴(2, 1); 𝐵(9, 1); 𝐶(9, 5)

𝐿𝑢𝑎𝑠 = 12±2 19 19 5

±

= 12[2 + 45 + 9 − (9 + 9 + 10)]

= 12|56 − 28| =

12. 28 = 14𝑠𝑎𝑡𝑢𝑎𝑛

2. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑙𝑢𝑎𝑠𝑑𝑎𝑒𝑟𝑎ℎ𝑦𝑔𝑑𝑖𝑏𝑎𝑡𝑎𝑠𝑖𝑜𝑙𝑒ℎ𝑡𝑖𝑡𝑖𝑘 𝐴(7, 16); 𝐵(12, 3); 𝐶(20, 7)

𝐽𝑎𝑤𝑎𝑏:

𝐿𝑢𝑎𝑠 = 12±7 89 312 7

±

= 12[21 + 63 + 96 − (72 + 36 + 49)]

= 12|180 − 157| =

12. |−23| = 11,5

3. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑙𝑢𝑎𝑠𝑑𝑎𝑒𝑟𝑎ℎ𝑦𝑔𝑑𝑖𝑏𝑎𝑡𝑎𝑠𝑖𝑜𝑙𝑒ℎ𝑡𝑖𝑡𝑖𝑘 𝐴(2,−1); 𝐵(7, 4); 𝐶(8, −1); 𝐷(1, 4)

𝑘𝑎𝑟𝑒𝑛𝑎𝑎𝑑𝑎 ≥ 3𝑡𝑖𝑡𝑖𝑘,𝑚𝑎𝑘𝑎ℎ𝑎𝑟𝑢𝑠𝑑𝑖𝑔𝑎𝑚𝑏𝑎𝑟𝑢𝑛𝑡𝑢𝑘𝑚𝑒𝑛𝑔𝑒𝑡𝑎ℎ𝑢𝑖𝑙𝑒𝑡𝑎𝑘𝑛𝑦𝑎, 𝑘𝑎𝑟𝑒𝑛𝑎𝑝𝑒𝑛𝑢𝑙𝑖𝑠𝑎𝑛𝑡𝑖𝑡𝑖𝑘𝑯𝑨𝑹𝑼𝑺𝑼𝑹𝑼𝑻,𝑠𝑒𝑎𝑟𝑎ℎ𝑗𝑎𝑟𝑢𝑚𝑗𝑎𝑚𝑎𝑡𝑎𝑢𝑏𝑒𝑟𝑙𝑎𝑤𝑎𝑛𝑎𝑛𝑗𝑎𝑟𝑢𝑚𝑗𝑎𝑚

𝑐𝑎𝑟𝑎𝑚𝑎𝑛𝑢𝑎𝑙:

𝐿𝑢𝑎𝑠 = 6 × 5 = 30𝑠𝑎𝑡𝑢𝑎𝑛(𝑝𝑎𝑛𝑗𝑎𝑛𝑔 × 𝑙𝑒𝑏𝑎𝑟)

𝑝𝑎𝑘𝑎𝑖𝑚𝑎𝑡𝑟𝑖𝑘𝑠:kalautidaksearahjarumjam:

𝐿𝑢𝑎𝑠 = 12Í𝐴𝐵𝐶𝐷

Í =12Í2 −17 41 48 −1

Í

= 12[8 + 28 − 1 − 8 − (−7 + 4 + 32 − 2)]

= 12[27 − 25] = 1𝑠𝑎𝑡𝑢𝑎𝑛? ? ? ? ?

searahjarumjam(clockwise):

𝐿𝑢𝑎𝑠 = 12Í𝐵𝐷𝐴𝐶

Í =12Í7 48 −12 −11 4

Í

= 12[−7 − 8 + 8 + 4 − (32 − 2 − 1 + 28)]

= 12|−3 − 57| = 30𝑠𝑎𝑡𝑢𝑎𝑛𝑜𝑘𝑒

berlawananarahjarumjam(anticlockwise):

𝐿𝑢𝑎𝑠 = 12Í𝐵𝐶𝐴𝐷

Í =12Í7 41 42 −18 −1

Í

= 12[28 − 1 − 2 + 32 − (4 + 8 − 8 − 7)]

= 12|57 − (−3)| = 30𝑠𝑎𝑡𝑢𝑎𝑛𝑜𝑘𝑒

12

Latihan 7:

1. 𝐷𝑒𝑛𝑔𝑎𝑛𝑐𝑎𝑟𝑎𝐼𝑛𝑣𝑒𝑟𝑠𝑚𝑎𝑡𝑟𝑖𝑘𝑠&𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑝𝑒𝑛𝑦𝑒𝑙𝑒𝑠𝑎𝑖𝑎𝑛𝑑𝑎𝑟𝑖 ∶

¹3𝑎 − 2𝑏 = 42𝑎 + 3𝑏 = 33

𝑗𝑎𝑤𝑎𝑏:

P Q P Q = P Q

↓ 𝑑𝑒𝑡 =

𝐼𝑛𝑣𝑒𝑟𝑠𝑚𝑎𝑡𝑟𝑖𝑘𝑠: 𝑐𝑎𝑟𝑎𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛:

𝑎 = _________

𝑏 = _________

2. 𝐷𝑒𝑛𝑔𝑎𝑛𝑐𝑎𝑟𝑎𝐼𝑛𝑣𝑒𝑟𝑠𝑚𝑎𝑡𝑟𝑖𝑘𝑠&𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛 𝑡𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑝𝑒𝑛𝑦𝑒𝑙𝑒𝑠𝑎𝑖𝑎𝑛𝑑𝑎𝑟𝑖 ∶

»4𝑥 + 3𝑦 = 30𝑥 + 4𝑦 = 1

𝑗𝑎𝑤𝑎𝑏:

P Q P Q = P Q

↓ 𝑑𝑒𝑡 =

𝐼𝑛𝑣𝑒𝑟𝑠𝑚𝑎𝑡𝑟𝑖𝑘𝑠: 𝑐𝑎𝑟𝑎𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛:

𝑥 = _________

𝑦 = _________

3. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑙𝑢𝑎𝑠𝑑𝑎𝑒𝑟𝑎ℎ𝑦𝑔𝑑𝑖𝑏𝑎𝑡𝑎𝑠𝑖𝑜𝑙𝑒ℎ𝑡𝑖𝑡𝑖𝑘 (4, 8); (3, 0); (−1, 2); (7, 6)

4. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑙𝑢𝑎𝑠𝑑𝑎𝑒𝑟𝑎ℎ𝑦𝑔𝑑𝑖𝑏𝑎𝑡𝑎𝑠𝑖𝑜𝑙𝑒ℎ𝑡𝑖𝑡𝑖𝑘 (4, 8); (3, 0); (−1, 1); (7, 2); (0, 6)

5. 𝑇𝑒𝑛𝑡𝑢𝑘𝑎𝑛𝑙𝑢𝑎𝑠𝑑𝑎𝑒𝑟𝑎ℎ𝑦𝑔𝑑𝑖𝑏𝑎𝑡𝑎𝑠𝑖𝑜𝑙𝑒ℎ𝑡𝑖𝑡𝑖𝑘 (6, 7); (3, 1); (0, 2); (0, 6); (2, 8)

EVALUASI𝑘𝑒𝑟𝑗𝑎𝑘𝑎𝑛𝑠𝑜𝑎𝑙ℎ𝑎𝑙𝑎𝑚𝑎𝑛98 − 100