operasi pada matriks

1. MTKU – 3.3/4.3/3/3.3
OPERASI PADA MATRIKS
1. Identitas
a. Nama Mata Pelajaran : MatematikaWajib
b. Semester : 3
c. Materi Pokok : Operasi pada Matriks
d. AlokasiWaktu : 4 JP x 2
e. Kompetensi Dasar :
3.3 Menjelaskan matriks dan kesamaan matriks dengan menggunakan masalah kontekstual dan
melakukan operasi pada matriks yang meliputi penjumlahan, pengurangan, perkalianskalar,
dan perkalian, serta transpose
4.3 Menyelesaikan masalah kontekstual yang berkaitan dengan matriks dan operasinya
f. Tujuan Pembelajaran:
Melalui Pendekatan saintifik dengan menggunakan model pembelajaran Problem Based Learning
dan metode diskusi, peserta didik dapat menganalisis konsep matriks dan operasi aljabar pada
matriks dengan mengembangkan sikap religius, mandiri, jujur, penuhtanggung jawab, teliti,
bekerja keras dan bekerja sama.
g. Materi Pembelajaran
1. Sinaga, Bornok, dkk. 2017. Buku Siswa Matematika XI. Jakarta: Kementrian Pendidikan dan
Kebudayaan
2. Sukino. 2017. Matematika Wajib 2A untuk SMA/MA kelas XI. Jakarta: Erlangga
1. Pastikan dan fokuskan apa yang akan anda pelajari hari ini.
2. Baca dan pahami Pendahuluan (Apersepsi) untuk membantu anda
memfokuskan permasalahan yang akan dipelajari.
3. Cari referensi/buku-buku teks yang terkait dengan
topik/permasalahan yang anda hadapi.
4. Jangan lupa browsing internet untuk menda-patkan pengetahuan
yang up to date.
5. Selalu diskusikan setiap persoalan yang ada dengan teman-teman
dan atau guru.
6. Presentasikan hasil pemahaman anda agar bermanfaat bagi orang
lain.
Jika tahapan-tahapan telah kalian lewati, kalian boleh meminta tes formatif kepada Bp/Ibu guru
sebagai prasyarat untuk melanjutkan ke UKBM berikutnya. Oke.?!
Petunjuk Umum

2. MTKU – 3.3/4.3/3/3.3
h. Kegiatan Pembelajaran
a) Pendahuluan
Sebelum belajar pada materi ini silahkan kalian membaca dan memahami uraian di bawah ini.
Seeorang wisatawan local hendak berlibur kebeberapa tempat
wisata yang ada di pulau Jawa. Untuk memaksimalkan waktu
liburan, dia mencatat jarak antara kota-kota tersebut sebagai
berikut.
Bandung – Semarang 367 km
Semarang – Yogyakarta 115 km
Bandung – Yogyakarta 428 km
Dapatkah kalian membuat susunan jarak antar kota tujuan wisata
tersebut jika wisatawan tersebut memulai perjalannya dari Bandung? Kemudian, berikan makna
setiap angka dalam susunan tersebut.
Nah, agar kalian dapat menyelesaikan permasalahan tersebut, marikita pelajari UKBM ini.
b) Peta Konsep

3. MTKU – 3.3/4.3/3/3.3
Pengertian dan Jenis-jenis Matriks
2. Kegiatan Inti
KegiatanBelajar 1
Baca dan pahamilah buku teks pelajaran halaman 74 – 84.
Agar kalian lebih memahami tentang matriks, perhatikanlah uraian berikut.
Amatilah permasalahan pada kegiatan pendahuluan. Dapatkah kalian menyelesaikan permasalahan tersebut?
Diskusikan dengan teman kalian bagaimana menyelesaikan permasalahan pada kegiatan pendahuluan.
Jika permasalahan pada kegiatan pendahuluan kalian nyatakan dalam bentuk tabel, maka akan kalian
peroleh:
Bandung Semarang Yogyakarta
Bandung 0 367 …
Semarang … ... 115
Yogyakarta … ... ...
Tabel di atas berisi baris dan kolom. Apabila dinyatakan dalam bentuk matriks, maka akan kalian peroleh
(
0 367 …
… … 115
… … …
)
Elemen matriks baris pertama kolom kedua (𝑎12) adalah … .
Elemen matriks baris kedua kolom ketiga (𝑎23) adalah … .
Matriks adalah
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Penamaan suatu matriks biasa menggunakan
huruf kapital.
𝐴 = (
𝑎11 𝑎12 …
𝑎21 𝑎22 …
⋮ ⋮ 𝑎𝑖𝑗
𝑎 𝑚1 𝑎 𝑚2 …
𝑎1𝑛
𝑎2𝑛
⋮
𝑎 𝑚𝑛
)
Matriks tersebut terdiri dari m baris dan n kolom.
Keterangan:
 𝑎12 adalah elemen baris ke-1 dan kolom
ke-2
 Elemen-elemen : 𝑎11 , 𝑎12 , … , 𝑎1𝑛 disebut
elemen-elemen penyusun baris 1
 Elemen-elemen : 𝑎13 , 𝑎23 , … , 𝑎 𝑚3
disebut elemen-elemen penyusun kolom
3
 Dan elemen𝑎𝑖𝑗 adalah elemen baris ke 𝑖
dan kolom ke 𝑗
𝑖 = 1, 2, 3, … , 𝑚
𝑗 = 1, 2, 3, … , 𝑛
Apa yang kalian ketahui tentang
matriks??
Suatu matriks A yang
memiliki 𝑚 baris dan 𝑛
kolom disebut matriks
berordo 𝑚 × 𝑛, dan diberi
notasi “𝐴 𝑚×𝑛”.

4. MTKU – 3.3/4.3/3/3.3
Kalau begitu apakah kalian sudah memahami konsep matriks?
Agar kalian lebih memahami konsep matriks cobalah selesaikan permasalahan pada kegiatan Ayo Berlatih
berikut.
Permasalahan 1
Suatu perusahaan pakaian, JCloth, memiliki dua pabrik yang terletak di Surabaya dan Malang. Di dua
pabrik tersebut, JCloth memproduksi dua jenis pakaian, yaitu kaos dan jaket. Perusahaan tersebut
memproduksi pakaian yang kualitasnya dapat dibedakan menjadi tiga jenis, yaitu standard, deluxe, dan
premium. Tahun kemarin, pabrik di Surabaya dapat memproduksi kaos sebanyak 3.820 kualitas
standard, 2.460 kualitas deluxe, dan 1.540 kualitas premium, serta jaket sebanyak 1.960 kualitas
standard, 1.240 kualitas deluxe, dan 920 kualitas premium. Sedangkan pabrik yang terletak di Malang
dapat memproduksi kaos sebanyak 4.220 kualitas standard, 2.960 kualitas deluxe, dan 1.640 kualitas
premium, serta jaket sebanyak 2.960 kualitas standard, 3.240 kualitas deluxe, dan 820 kualitas
premium dalam periode yang sama.
Bagaimanakah bentuk matriks produksi dari permasalahan di atas?
Jawaban
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Apakah kalian sudah mengetahui tentang jenis-jenis matriks? Jika belum baca dan pahamilah materi pada
buku paket halaman 80 – 83 tentang jenis-jenis matriks, kemudian coba kalian lengkapi tabel di bawah ini.
No Jenis
Matriks
Contoh Banyaknya
Baris (m)
Banyaknya
Kolom (n)
Ordo
(m x n)
1. Matriks
Baris
𝐴 = (1 3 6) 1 3 1 x 3
2. Matriks
Kolom
𝐵 = (
−1
0
) … … …
3. Matriks
Persegi
Panjang
… … …
3 x 2
4. Matriks
Persegi … … …
3 x 3
5. Matriks
Segitiga …
3
… …
6. Matriks
Diagonal … …
3
…
7. Matriks
Identitas … … … …
8. Matriks Nol
… … … …
AYO BERLATIH

5. MTKU – 3.3/4.3/3/3.3
Kesamaan Dua Matriks Dan Transpose Matriks
Setelah kalian memahami tentang pengertian matriks , coba kalian pelajari tentang Kesamaan Dua Matriks
pada buku teks pelajaran halaman 84 – 85 dan materi transpose matriks pada halaman 98. Selanjutnya
cobalah selesaikan permasalahan berikut.
Periksa kesamaan matriks berikut:
a. 𝐴 = (
3 −1
√9 2
) dan 𝐵 = (
6
2
−1
3 √4
)
b. 𝐶 = (
2 −1
3 4
) dan 𝐷 = (
2 4
−1 3
)
Penyelesaian:
a. Matriks A dan B berordo sama yaitu 2 x 2 dan elemen-elemen matriks yang seletak ............. Sehingga
matriks A = B.
b. Matriks C dan D berordo sama yaitu 2 x 2, tetapi elemen-elemen yang seletak ....................... sehingga C
tidak sama dengan D.
Agar kalian lebih memahami materi kesamaan dua matriks, kerjakan soal pada Ayoo Berlatih berikut.
1. Jika matriks 𝐴 = (
2 4
5 3
) sama dengan 𝐵 = (
2 𝑥
𝑦 3
), maka tentukan nilai 𝑥 + 𝑦.
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2. Diberikan matriks 𝐴 = (
2𝑥 − 𝑦 −3
−4 −8
) dan matriks 𝐵 = (
−2 −3
−4 𝑥 + 𝑦
). Jika 𝐴 = 𝐵 maka tentukan nilai
dari 2𝑥 + 𝑦.
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3. Jika 𝐴 = (
2 4
5 3
) maka tentukan transpose matriks A (𝐴𝑡
).
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AYO BERLATIH

6. MTKU – 3.3/4.3/3/3.3
Operasi Penjumlahan dan Pengurangan Pada Matriks
Kegiatan Belajar 2
4. Mungkinkah suatu matriks sama dengan transpose matriksnya sendiri? Jelaskan dan berikan contoh.
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Apabila kalian telah mampu menyelesaikan persoalan di atas, maka kalian bisa melanjutkan pada kegiatan
belajar 2 berikut.
Untuk memahami materi operasi penjumlahan dan pengurangan matriks, bacalah materi pada buku teks
pelajaran hal 86 – 92.
Setelah kalian membaca materi tersebut, apakah yang kalian ketahui tentang penjumlahan dan pengurangan
matriks?
Contoh:
Jika diketahui𝐴 = (
2 −5 7
4 2 −1
), 𝐵 = (
0 3
2 2
−1 0
), dan𝐶 = (
−2 5 −7
−4 −2 1
), tentukan:
a. 𝐴 + 𝐶
b. 𝐴 − 𝐶
c. 𝐴 + 𝐵
d. 𝐵 − 𝐶
Penyelesaian:
a. 𝐴 + 𝐶 = (
2 −5 7
4 2 −1
) + (
−2 5 −7
−4 −2 1
)
= (
2 + ⋯ −5 + ⋯ …+ (−7)
… + (−4) 2 + (… ) −1 + ⋯
)
= (
… 0 …
0 … 0
)
b. 𝐴 − 𝐶 = (
2 −5 7
4 2 −1
) − (
−2 5 −7
−4 −2 1
)
= (
2 − (… ) −5 − ⋯ 7 − (… )
… − (−4) …− (−2) … − 1
)
= (
4 −10 …
… … −2
)
c. 𝐴 + 𝐵 = .......................... sebab ordo matriks A ≠ ordo matriks B
d. 𝐵 − 𝐶 = .......................... sebab ordo matriks B ≠ ordo matriks C
Nah, agar kalian lebih memahami penjumlahan dan pengurangan matriks, kerjakan soal pada kegiatan Ayoo
Berlatih berikut.
Misalkan A dan B adalah matriks berordo m x n dengan entry-entry 𝑎 𝑖𝑗dan 𝑏𝑖𝑗. Matriks C
adalah.............. matriks A dan B, ditulis 𝐶 = 𝐴 + 𝐵, apabila C juga berordo m x n dengan
entry-entry ditentukan oleh : 𝑐𝑖𝑗 = 𝑎 𝑖𝑗 + 𝑏𝑖𝑗 (untuk semua 𝑖dan𝑗).
Sedangkan pengurangan matriks A dengan matriks B didefinisikan sebagai ............. antara
matriks A dan – B, dengan matriks – B adalah lawan matriks B sehingga:
𝐴 − 𝐵 = 𝐴 + (−𝐵)

7. MTKU – 3.3/4.3/3/3.3
1. Jika matriks 𝐴 = (
4 −10
12 11
)dan𝐵 = (
2 −5
4 2
), maka tentukan:
a. 𝐴 + 𝐵
b. 𝐵 + 𝐴
c. Kesimpulan dari a dan b
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2. Jika matriks 𝐴 = (
2 1
4 5
), 𝐵 = (
−3 2
1 4
), 𝑑𝑎𝑛 𝐶 = (
5 1
4 3
), maka tentukanlah:
a. (A + B) + C
b. A + (B + C)
c. Kesimpulan dari a dan b
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3. Jika matriks 𝐴 = (
2 1
4 5
)dan 𝑂 = (
0 0
0 0
), maka tentukan:
a. A + O
b. O + A
c. Kesimpulan dari a dan b
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AYO BERLATIH

8. MTKU – 3.3/4.3/3/3.3
4. Tentukan nilai 𝑥 dan 𝑦 yang memenuhi persamaan
(
3 3𝑥
−1 𝑦
) − (
2 𝑦 + 𝑥
𝑦 + 1 𝑥
) = (
1 3
3 −4
)
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5. Tentukan matriks P dari operasi matriks 𝑃 + (
2 −1
0 3
) = (
4 −2
3 5
).
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6. Perhatikan Permasalahan 1 pada Kegiatan belajar 1, tentukan banyaknya pakaian yang telah diproduksi
di pabrik Surabaya dan Malang.
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Sifat-sifat penjumlahan matriks:
1. …………………………………
2. …………………………………
3. …………………………………
4. …………………………………

9. MTKU – 3.3/4.3/3/3.3
Operasi Perkalian Matriks
Sebelum mempelajari materi operasi matriks pada UKBM ini bacalah terlebih dahulu buku teks pelajaran
halaman 92 – 99.
Setelah kalian membaca materi perklian matriks pada buku, dan memahami sifat-sifatnya, selanjutnya
kerjakan soal pada kegiatan Ayoo Berlatih berikut ini.
1. Tentukan matriks yang diwakili oleh:
a. 3 (
2 −1
3 2
)
b. 4 (
1 0
2 −3
) − 3 (
3 −2
4 0
)
c. (5 + 3)(
4 −1
3 2
)
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2. Tentukan nilai 𝑎, 𝑏, 𝑐 yang memenuhi persamaan
𝑎 (
3
−3
0
) + 𝑏 (
−1
5
2
) + 𝑐 (
4
1
−5
) = (
11
−22
−11
)
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Sifat-sifatPerkalianMatriks
1. Jika 𝑘, 𝑙 ∈ 𝑅, matriks 𝐴 = (𝑎𝑖𝑗 ) berordo m x n dan 𝐵 = (𝑏𝑖𝑗)berordo n x m, maka:
a. ( 𝑘 + 𝑙) 𝐴 = 𝑘𝐴 + 𝑙 … dan ( 𝑘 − 𝑙) 𝐴 = ⋯ 𝐴 − 𝑙𝐴
b. 𝑘( 𝐵𝐴) = ( 𝑘 … ) 𝐴
c. 𝑘(𝑙𝐴) = (… 𝑙) 𝐴
2. 𝐴𝐵 ≠ 𝐵𝐴, yaitu tidak berlaku sifat komutatif
3. 𝑈𝑛𝑡𝑢𝑘 𝑠𝑒𝑚𝑏𝑎𝑟𝑎𝑛𝑔 k ∈ R, 𝐴 = (𝑎𝑖𝑗 ), dan 𝐵 = (𝑏𝑖𝑗), maka:
a. ( 𝑘𝐴) 𝐵 = ⋯ ( 𝐴𝐵)
b. ( 𝐴𝑘) 𝐵 = ⋯( 𝑘𝐵)
c. ( 𝐴𝐵) 𝑘 = 𝐴(… 𝑘)
4. Untuk 𝐴 = ( 𝑎𝑖𝑗 ), 𝐵 = ( 𝑏𝑖𝑗), 𝑑𝑎𝑛 𝐶 = (𝐶𝑖𝑗), maka:
a. 𝐴( 𝐵𝐶) = ( 𝐴𝐵) 𝐶, jika 𝐴𝐵 𝑑𝑎𝑛 𝐵𝐶 terdefinisi, atau memenuhi sifat ..................
b. 𝐴( 𝐵 + 𝐶) = 𝐴𝐵 + 𝐴𝐶, jika𝐴𝐵, 𝐴𝐶, dan 𝐵 + 𝐶 terdefinisikan. Sifat ini disebut
........................ kiri terhadap penjumlahan.
c. ( 𝐴 + 𝐵) 𝐶 = 𝐴𝐶 + 𝐴𝐵, jika 𝐴𝐶, 𝐵𝐶, 𝑑𝑎𝑛 (𝐴 + 𝐵) terdefinisi. Sifatinidisebut
distributive kananterhadappenjumlahan.
AYO BERLATIH
Perkalian matriks dengan matriks hanya dapat
dikalikan apabila …..

10. MTKU – 3.3/4.3/3/3.3
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3. Jika 𝐴 = (
−1 2
−3 1
) 𝑑𝑎𝑛 𝐵 = (
0 −1
3 2
), tentukan:
a. 𝐴2
b. 𝐵2
c. ( 𝐴𝐵)2
d. 𝐴2
𝐵2
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4. Jumlah rata-rata roti tawar, roti keju, dan roti coklat yang diproduksi oleh dua pabrik roti ditunjukkan
oleh tabel berikut.
Jumlah roti tawar yang diproduksi hari Senin:
Roti tawar Roti keju Roti Cokelat
Pabrik P 840 320 360
Pabrik Q 410 580 275
Jumlah roti tawar yang diproduksi hari Kamis:
Roti tawar Roti keju Roti Cokelat
Pabrik P 250 910 625
Pabrik Q 435 825 500
Dari permasalahan di atas, tentukan:
a. Model matriks dari masalah tersebut
b. Jumlah produksi keseluruhan masing-masing jenis roti pada hari Senin dan Kamis
c. Apabila jumlah produksi roti pada hari Senin ditingkatkan menjadi 4 kali lipat dan jumlah produksi
roti pada hari Kamis ditingkatkan 3 kali lipat, tentukan jumlah produksi masing-masing jenis roti
pada kedua hari tersebut.
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5. Tabel berikut menunjukkan jumlah barang-barang yang telah dibuat oleh 4 orang pegawai suatu pabrik
furniture pada suatu hari.
Pegawai Kursi Meja Lemari Upah per
Hari
Ahmad 5 2 1 Rp150.000,00
Budi 4 12 0 Rp160.000,00
Casmat 5 4 1 Rp165.000,00
Dadang 10 0 3 Rp175.000,00
Bonus untuk pembuatan masing-masing barang adalah sebagai berikut.
Kursi : Rp5.000,00/unit

11. MTKU – 3.3/4.3/3/3.3
Meja : Rp10.000,00/unit
Lemari : Rp15.000,00/unit
Tentukan, berapakah upah dan bonus yang diterima masing-masing pegawai pada hari itu.
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12. MTKU – 3.3/4.3/3/3.3
3. Penutup
Setelah kalian belajar bertahap dan berlanjut melalui kegiatan belajar 1, dan 2, berikut untuk mengukur diri
kalian terhadap materi yang sudah kalian pelajari. Jawablah sejujurnya terkait dengan penguasaan materi
pada UKBM ini di Tabel berikut.
Tabel Refleksi Diri Pemahaman Materi
No Pertanyaan Ya Tidak
1. Apakah Anda dapat menjelaskan tentang pengertian
matris, notasi, dan ordo suatu matriks?
2. Apakah Anda dapat menjelaskan tentang kesamaan
dua matriks dan transpose matriks?
3. Apakah Anda dapat menjelaskan tentang operasi
penjumlahan, pengurangan, dan perkalian pada
matriks?
4. Apakah Anda dapat menyelesaikan masalah
kontekstual berkaitan dengan penjumlahan dan
pengurangan menggunakan konsep matriks?
5. Apakah Anda dapat menyelesaikan masalah
kontekstual berkaitan dengan perkalian
menggunakan konsep matriks?
Jika menjawab “TIDAK” pada salah satu pertanyaan di atas, maka pelajarilah kembali materi tersebut dalam
Buku Teks Pelajaran (BTP) dan pelajari ulang UKBM ini dengan bimbingan Guru atau teman sejawat.
Jangan putus asa untuk mengulang lagi!. Dan apabila kalian menjawab “YA” pada semua pertanyaan,
maka kalian boleh sendiri atau mengajak teman lain yang sudah siap untuk mengikuti tes formatif agar
kalian dapat belajar ke UKBM berikutnya... Oke.?
Referensi:
- Sinaga, Bornok, dkk. 2017. Buku Siswa Matematika XI. Jakarta: Kementrian Pendidikan dan
Kebudayaan
- Sukino.Edisi revisi 2017.Matematika Wajib 2A untuk SMA/MA kelas XI. Jakarta: Erlangga