Tugas 2.4. LKPD - Bp. Kusno - Desy Sofia [PDF]

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LEMBAR KERJA PESERTA DIDIK (LKPD)



Satuan Pendidikan : SMA Negeri 1 Sokaraja Mata Pelajaran



: Matematika Wajib



Kelas / Semester



: XI / Ganjil



Materi



: Persamaan Matriks berbentuk AX=B dan XA=B



A. Identitas Kelompok : ………………………………………………….. Kelas : ………………………………………………….. Angggota Kelompok : 1. ……………………………………………….. 2. ……………………………………………….. 3. ……………………………………………….. 4. ……………………………………………….. 5. ……………………………………………….. B. Tujuan Pembelajaran



1. Dengan



menggunakan



matriks



nonsingular



ordo



2x2,



peserta



didik



dapat



mengidentifikasi sifat invers matriks secara tepat. 2. Dengan menggunakan sifat invers matriks dan sifat matriks identitas, peserta didik dapat menemukan konsep persamaan matriks berbentuk AX=B dan XA=B dengan pengetahuan invers matriks yang dimilikinya. 3. Dengan konsep persamaan matriks yang telah ditemukan, peserta didik dapat menentukan penyelesaian dari persamaan matriks berbentuk AX=B dan XA=B dengan benar. 4. Dengan konsep penyelesaian persamaan matriks AX=B, peserta didik dapat menyelesaikan model matematika dari suatu masalah nyata yang berkaitan dengan persamaan matriks metode invers matriks secara tepat.



C. Petunjuk 1.



Berdoalah terlebih dahulu sebelum mengerjakan LKPD.



2.



Kerjakan LKPD dengan cara berkelompok.



3.



Tuliskan jawaban kalian pada tempat yang telah disediakan pada masing-masing kegiatan.



4.



Kerjakanlah dengan teliti dan urut.



1



D.Uraian Materi



Materi Prasayat Peserta didik harus sudah menguasai konsep invers matriks ordo 2x2



Invers Matriks ordo 2x2



a b  Diketahui matriks A   maka invers matriks A adalah A1 .  c d  1 A1   Adjoin A det A 1  d b  dengan ad  bc  0 .  ad  bc  c a 



Kegiatan 1



Mengidentifikasi Salah Satu Sifat Invers Matriks



Petunjuk : Jawablah pertanyaan pada kegiatan 1 ini untuk mengidentifikasi salah satu sifat invers matriks.



1 2  Diberikan matriks A   . 1 3   Carilah invers matriks A : ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ...............................................................................................................................................  Kalikan matriks A dengan invers matriks A : a.



A  A1 ........................................................................................................................................ ........................................................................................................................................ ........................................................................................................................................ ........................................................................................................................................ Matriks hasil perkalian A  A1 disebut Matriks .......................................................... 2



b. A1  A ........................................................................................................................................ ........................................................................................................................................ ........................................................................................................................................ ........................................................................................................................................ Matriks hasil perkalian A1  A disebut Matriks ..........................................................



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Coba simpulkan sifat invers mastriks yang kalian temukan dari hasil identifikasi di atas ! ........................................................................................................... ........................................................................................................... ........................................................................................................... ........................................................................................................... ........................................................................................................... ...........................................................................................................



3



Menemukan konsep persamaan matriks bentuk AX=B dan XA = B



Kegiatan 2



Petunjuk : Isilah titik-titik pada kegiatan 2 ini dengan sifat invers matriks yang telah kalian temukan pada kegiatan 1 dan pengetahuan matriks yang sudah kalian miliki. a. Persamaan Matriks Berbentuk AX=B Diketahui matriks A nonsingular, mari temukan cara mencari matriks X !  Dari persamaan matriks AX=B, kalikan kedua ruas dengan A1 dari kiri AX  B ...  A  X  ...  B Cari informasi yuuuk… .....  ..... Bagaimana sifat matriks .....  ..... identitas (I)..? X  .....



. b. Persamaan Matriks Berbentuk XA=B Diketahui matriks A nonsingular, mari temukan cara mencari matriks X !  Dari persamaan matriks XA=B, kalikan kedua ruas dengan A1 dari kanan. XA  B X  A  ...  B  ... .....  ..... .....  ..... X  ..... 2



Dari penemuan konsep pada kegiatan 2, maka dapat disimpulkan penyelesaian persamaan matriks berbentuk : a.



AX  B  X  ........



b. XA  B  X  ........



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Menyelesaikan Masalah yang Berkaitan dengan Persamaan Matriks Kegiatan 3



Dodi dan Yanti pergi ke supermarket. Disana, Dodi membeli 4 bungkus roti dan 3 botol minuman, sedangkan Yanti hanya membeli 2 bungkus roti dan sebotol minuman dengan jenis dan merk yang sama dengan yang dibeli Doni. Jika Dodi harus membayar Rp.44.800,00 dan Yanti membayar Rp.20.000, tentukkan harga 1 bungkus roti dan 1 botol minuman dengan metode invers matriks ! Sumber: Surya-Tribunnews



Penyelesaian :  Misalkan : x = ……………………… y = ……………………… …



…



(x)



(y)



Doni



…



…



…



Yanti



…



…



…



Harga



...x  ... y  ......... ...x  ... y  .........



 Nyatakan permasalahan tersebut ke dalam sistem persamaan linier 



 Susunlah sistem persamaan linier diatas dalam bentuk matriks :



... ...  x  ......... ... ...  y   .........     



 Disusun dalam bentuk persamaan matriks, sehingga :



... ...  x  ......... ... ...  y   .........      A X  B  Berdasarkan konsep persamaan matriks AX=B, maka X = ……..



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 Carilah matriks X menggunakan konsep yang telah kalian temukan dan pengetahuan tentang matriks yang telah kalian ketahui. X  ................................................. ....  ................................................. ....  ................................................. ....  ................................................. ....  .................................................



 Ambil kesimpulan kalian untuk permasalahan diatas, Jadi harga 1 bungkus roti adalah ……….. dan harga 1 botol minuman adalah ………….



E. Latihan Kerjakan soal-soal berikut ini dengan konsep yang telah kalian temukan !



1 3  4 5 1. Tentukanlah matriks X ordo 2x2 yang memenuhi persamaan X    !  2 1   4 2 1 3   x   0  2. Dengan metode invers matriks, tentukan nilai x dan y dari persamaan       !  2 4  y   2



 4 1  4 50   2 4 3. Diketahui matriks A   , B dan C    .   3 5   2 60   1 3 Tentukan matriks M ordo 2x2 yang memenuhi persamaan AMB  C ! 4. Sebuah tangki A berisi campuran 20 liter air dan 10 liter alkohol. Tangki B berisi campuran 24 liter air dan 6 liter alkohol. Dengan metode invers matriks, berapa liter harus diambil dari tiap-tiap tangki agar diperoleh campuran sebanyak 16 liter dengan kadar alkohol 25%?



Jawab: ...................................................................................................................................................... ...................................................................................................................................................... 6



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