Misalkan ada matriks A, maka invers matriks A dapat dituliskan A-1 dan memenuhi sifat:
A x A-1 = A-1 x A = I
I adalah matriks identitas. Untuk matriks berordo 2 x 2 nilai I adalah ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqAYTSUC_N90eu56GVyA88HXuQsYizakgw7jedTVeKmEYH-NOoW_iGExMWtfYI4hKS_ypAaAQN0M_fAldoFv02K6gfJVF_WiILgauxcXaLj0BRdzlHJdW5WHkf0B0iBMjuE01Lygtrzwo/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
Syarat matriks memiliki invers:
1. Jika ber ordo n x n dan determinannya tidak sama dengan nol.
2. Matriks A disebut matriks non singular atau memiliki invers jika det A ≠ 0
3. Matriks A disebut matriks singular atau tidak memiliki invers jika det A = 0
Rumus invers matriks:
Sifat invers pada matriks adalah:
1. (A-1)-1 = A
2. (A . B)-1 = A-1 . B-1
3. (AT)-1 = (A-1)T
4. A-1 . A = I
Matriks A dan B dikatakan saling invers jika AB = I dan BA = I.
“Kak... aku bingung.....” oke, don’t worry.... kakak akan bagikan contohnya.
1. Diketahui matriks A =
dan matriks B =
. Apakah matriks A dan B saling invers?
Jawab:
A dan B dikatakan saling invers jika hasil dari AB = I = ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEioYmoSkO4jom0354gxn3yF0Qla9UBp61MM3EQ-F075jK2dgHnfTpd15egAuGUAtrleEgUYJq_MvIpg-ysG7ZYQrbu6vfRyRFrbCdrX4bE0ElUVnBJCUTkhpXk-7lFHO4CMuGoigVma3fc/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
Karena nilai A . B = I =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEioYmoSkO4jom0354gxn3yF0Qla9UBp61MM3EQ-F075jK2dgHnfTpd15egAuGUAtrleEgUYJq_MvIpg-ysG7ZYQrbu6vfRyRFrbCdrX4bE0ElUVnBJCUTkhpXk-7lFHO4CMuGoigVma3fc/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
maka matriks A dan B saling invers.
2. Dari matriks-matriks berikut ini, manakah yang memiliki invers?
Matriks A =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2OWYIgJXotwbJP4l0pC5ZvfR8eaNpJQl3s5GPk9r8Io2QpVxJYzA2fMy-EAYC7dDhzwDv3ToQHdEVwyWs6Jn6lYRwcyctQLkMBgbh8AEUQJzlNC4nycyfaG549F3z2Rk684poQ5TgRMY/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
; matriks B =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggBVhdqVrOru86K0AD43EFCsS31U7n5CZGAL_VQ3iwtyS2_hfPP3iTPvcoSRVfwOH6aVPgKivKGfyEBmV1X-cRD3uDbXlwIoSOb8u100STFqGQXisY3Wmpy6HV6uG2jn-D_Tdbu_BlHr8/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
; dan matriks C =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiX8NkVM5XoamRzBOZtN4gff9fXdq9taSx-c4txWjRIs57DCClXSn4HSeV9AQMcdKadNujYtrR00RWKdo5gAFt8cAgz2VwG_LtAer-6Kw0XW-ngNIEOFstYj2Vj-GGzskRLYrZ6_9so6Y/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
Jawab:
Matriks memiliki invers jika nilai determinannya tidak sama dengan 0.
Mari kita uji determinan matriks A, B, dan C di atas.
A =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2OWYIgJXotwbJP4l0pC5ZvfR8eaNpJQl3s5GPk9r8Io2QpVxJYzA2fMy-EAYC7dDhzwDv3ToQHdEVwyWs6Jn6lYRwcyctQLkMBgbh8AEUQJzlNC4nycyfaG549F3z2Rk684poQ5TgRMY/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
det A = (2
x 3) – (1 x 5) = 6 – 5 = 1
B =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggBVhdqVrOru86K0AD43EFCsS31U7n5CZGAL_VQ3iwtyS2_hfPP3iTPvcoSRVfwOH6aVPgKivKGfyEBmV1X-cRD3uDbXlwIoSOb8u100STFqGQXisY3Wmpy6HV6uG2jn-D_Tdbu_BlHr8/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
Det B = (-4 x 1) – (2 x 0) = -4 – 0 = -4
C =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiX8NkVM5XoamRzBOZtN4gff9fXdq9taSx-c4txWjRIs57DCClXSn4HSeV9AQMcdKadNujYtrR00RWKdo5gAFt8cAgz2VwG_LtAer-6Kw0XW-ngNIEOFstYj2Vj-GGzskRLYrZ6_9so6Y/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
Det C = (6 x -3) – (2 x -9) = -18 – (-18) = 0
Jadi, matriks A dan B memiliki invers, matriks C tidak memiliki invers.
3.
Tentukan invers dari matriks P =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNvfOwPObzJwA-2uNdLHXErKHEseUIMIaFKZ2m_Kcg4_752JJQiwsh_SIEMWsItWbftLNQDBcCcHYYusgCHkJzYwATeMIEYc5TuaDQP0tevjKOF3INOPYDttaA9YZPImqnq3bwr55Ohts/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
Jawab:
4.
Diketahui matriks A =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNAKbxwhZrKsH-6OixY1qC6HPQyDmRFECaL7UpoqkU_B8r6NNBzCK2xrr82-YYMsrXgj04_K9JtVEYsrHMrivTx7vygJ6rxn8XkxUck-SpS6o9crwOux-O3KlT-Gs12aypRbBVLzAygNo/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
dan matriks B =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNAcBazSwAwqHsOhUWRQL5DnsaSULEHoJzO90kw1hsMNvsD_zlIyAfAxTFlAvKsUQ4lAns_sDwFX4JUjbf7D4bxaQg6fSMRnAXbDyRqNRZKCKWeoTeggJcxtsmfVLDP7XxEXNPLE7-nBo/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
. Tentukan:
a. (AB)-1
b. B-1 . A-1
Jawab:
a. (AB)-1
b. B-1 . A-1
Berdasarkan sifat invers matriks,
B-1 . A-1 =
= (BA)-1 = (AB)-1 =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhP9fBYGZgiAr12DGuT6bhDUdlnrsAlNVHbpgyiNKH0qFlKskXkVMV_L2SDr4SY45jN26Dko0Mq3L-Oqop1EX_DkCKF3LA_0aoB9eFByoCdhpERFI4jWhJyZGKDL7cLziXp_oDEsfySSes/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
5.
Diketahui matriks A =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0UM8UzuccrLA27v2Rn8NOswq3po-l7fUj0J12kE4bR_f5HvFw7O9zxB1itY8mANkcByfU8xB0Xlw5CHCpmcbCkjOywIFe8Xdf4CieY5A6Tj4iikUVvgEEuhWYBXxzEJYWKIKLY9_45OI/s320/cara+mencari+invers+matriks+ordo+2+x+2.png)
. Tentukan nilai k yang memenuhi A + B =
C-1 !
Jawab:
Pertama kita cari C-1
Lalu, hitung A + B = C-1
-2 + (3k + 1) = 2
3k – 1 = 2
3k = 2 + 1
3k = 3
k = 3/3
k = 1
6.
Diketahui matriks A =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTSf5jTBfv-hUMDMDlbShYnIBmw0fBxGS-eELsUVBfDWSlgFmSZRU0XytCXumi9beDELscz2yF8NyZl25D_6pLL5ZEOdUNgbws79jb3TRsJGYJVSE_Yno698sYC4qFjuO-yyWyGw0j1G4/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
adalah invers matriks A =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFLLrecbSqLr00zxkgnOIop8PTDHZzAnCkZocraTczjyB-v50axR6URtdQAnEbKV8Ny4fysJGrYoBF9pLWCt-HMVEvZiczjRnsMNwBXN5AWMUjuEaIG82osKFFOZM7IFACV0io0ISlSpc/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
. Tentukanlah nilai x dan y yang memenuhi!
Jawab:
Karena nilai invers matriks A =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFLLrecbSqLr00zxkgnOIop8PTDHZzAnCkZocraTczjyB-v50axR6URtdQAnEbKV8Ny4fysJGrYoBF9pLWCt-HMVEvZiczjRnsMNwBXN5AWMUjuEaIG82osKFFOZM7IFACV0io0ISlSpc/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
maka:
x + 4 = 3
x = 3 – 4
x = -1
dan
2x + y = 3
2(-1) + y = 3
-2 + y = 3
y = 3 + 2
y = 5
Jadi, nilai x = -1 dan y = 5.
7.
Diketahui matriks P =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmQIhzr_5BaDXkZpmeCyoldonrX-ygvferoOJjbGTjZdzMveKB54Vx8yPYL8GLMKCPRrNbKHiHeXg9WCsxlSUXz3TqO6qmCkjWz6yKPWo1UcIYIndC2RrGgvc_jUTIV-TpJqkkJWeuV80/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
dan Q =
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmnLbgr38cK36BZpU2LwQ4Ta-J_lHMwhjLxyCpSv_viW-hxShvRb9Pc_w8GJKUE7BY7Kfd28wE1mW9C-YqkZ3qeZLYsIW4vCStIqY6E3t2K1NDY3VBxH8p-LtIWO16D-GNKNj88Hp7X0w/s0/cara+mencari+invers+matriks+ordo+2+x+2.png)
. Nilai determinan matriks
Q-1 . P-1 adalah...
Jawab:
Berdasarkan sifat invers matriks, nilai Q-1 . P-1
= (QP)-1
Sekian dulu ya belajar bersama ajar hitung hari ini.. sampai bertemu di materi selanjutnya...
No comments:
Post a Comment