Jika A dan B adalah matriks persegi, dan berlaku alt="A \cdot B = B \cdot A = I" class="latex" src="http://s.wordpress.com/latex.php?latex=A%20%5Ccdot%20B%20%3D%20B%20%5Ccdot%20A%20%3D%20I&bg=T&fg=000000&s=0" style="border:none; max-width:100%; vertical-align:middle" title="A \cdot B = B \cdot A = I" /> maka dikatakan matriks A dan B saling invers. B disebut invers dari A, atau ditulis alt="A^{-1}" class="latex" src="http://s.wordpress.com/latex.php?latex=A%5E%7B-1%7D&bg=T&fg=000000&s=0" style="border:none; max-width:100%; vertical-align:middle" title="A^{-1}" /> . Matriks yang mempunyai invers disebut invertible atau matriks non singular, sedangkan matriks yang tidak mempunyai invers disebut matriks singular.
Untuk mencari invers matriks persegi berordo 2×2, coba perhatikan berikut ini.
Jika alt="A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}" class="latex" src="http://s.wordpress.com/latex.php?latex=A%20%3D%20%5Cbegin%7Bbmatrix%7D%20a%20%26%20b%20%5C%5C%20c%20%26%20d%20%5Cend%7Bbmatrix%7D&bg=T&fg=000000&s=0" style="border:none; max-width:100%; vertical-align:middle" title="A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}" /> dengan alt="ad - bc \neq 0" class="latex" src="http://s.wordpress.com/latex.php?latex=ad%20-%20bc%20%5Cneq%200&bg=T&fg=000000&s=0" style="border:none; max-width:100%; vertical-align:middle" title="ad - bc \neq 0" /> , maka invers dari matriks A (ditulis alt="A^{-1}" class="latex" src="http://s.wordpress.com/latex.php?latex=A%5E%7B-1%7D&bg=T&fg=000000&s=0" style="border:none; max-width:100%; vertical-align:middle" title="A^{-1}" /> ) adalah sebagai berikut:
Jika
Jika alt="ad - bc = 0" class="latex" src="http://s.wordpress.com/latex.php?latex=ad%20-%20bc%20%3D%200&bg=T&fg=000000&s=0" style="border:none; max-width:100%; vertical-align:middle" title="ad - bc = 0" /> maka matriks tersebut tidak mempunyai invers, atau disebut matriks singular.
Sifat-sifat matriks persegi yang mempunyai invers:
alt="(A \cdot B)^{-1} = B^{-1} \cdot A^{-1}" class="latex" src="http://s.wordpress.com/latex.php?latex=%28A%20%5Ccdot%20B%29%5E%7B-1%7D%20%3D%20B%5E%7B-1%7D%20%5Ccdot%20A%5E%7B-1%7D&bg=T&fg=000000&s=0" style="border:none; max-width:100%; vertical-align:middle" title="(A \cdot B)^{-1} = B^{-1} \cdot A^{-1}" /> alt="(B \cdot A)^{-1} = A^{-1} \cdot B^{-1}" class="latex" src="http://s.wordpress.com/latex.php?latex=%28B%20%5Ccdot%20A%29%5E%7B-1%7D%20%3D%20A%5E%7B-1%7D%20%5Ccdot%20B%5E%7B-1%7D&bg=T&fg=000000&s=0" style="border:none; max-width:100%; vertical-align:middle" title="(B \cdot A)^{-1} = A^{-1} \cdot B^{-1}" /> alt="(A^{-1})^t =(A^{t})^{-1}" class="latex" src="http://s.wordpress.com/latex.php?latex=%28A%5E%7B-1%7D%29%5Et%20%3D%28A%5E%7Bt%7D%29%5E%7B-1%7D&bg=T&fg=000000&s=0" style="border:none; max-width:100%; vertical-align:middle" title="(A^{-1})^t =(A^{t})^{-1}" />
Contoh: Diketahui A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%202%20&%201%5C%5C%203%20&%202%20%5Cend%7Bbmatrix%7D" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; line-height:25px; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." /> dan B = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%202%20&%20-1%5C%5C%20-3%20&%202%20%5Cend%7Bbmatrix%7D" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; line-height:25px; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." />
Selidiki, apakah A dan B saling invers?
Penyelesaian :
Matriks A dan B saling invers jika berlaku A × B = B × A = I.
A × B = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%202%20&%201%5C%5C%203%20&%202%20%5Cend%7Bbmatrix%7D%5Cbegin%7Bbmatrix%7D%202%20&%20-1%5C%5C%20-3%20&%202%20%5Cend%7Bbmatrix%7D=%5Cbegin%7Bbmatrix%7D%201%20&%200%5C%5C%200%20&%201%20%5Cend%7Bbmatrix%7D=I" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." />
B × A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%202%20&%20-1%5C%5C%20-3%20&%202%20%5Cend%7Bbmatrix%7D%5Cbegin%7Bbmatrix%7D%202%20&%201%5C%5C%203%20&%202%20%5Cend%7Bbmatrix%7D=%5Cbegin%7Bbmatrix%7D%201%20&%200%5C%5C%200%20&%201%20%5Cend%7Bbmatrix%7D=I" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." />
Karena A × B = B × A maka A dan B saling invers, dengan A–1 = B dan B–1 = A.
Menentukan Invers Matriks 2 Berordo 2x2
Misalkan diketahui matriks A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%20a%20&%20b%5C%5C%20c%20&%20d%20%5Cend%7Bbmatrix%7D" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." /> , dengan ad – bc ≠ 0.
Suatu matriks lain, misalnya B dikatakan sebagai invers matriks A jika AB = I. Matriks invers dari A ditulis A–1 . Dengan demikian, berlaku :
AA–1 = A–1A = I
Matriks A mempunyai invers jika A adalah matriks nonsingular, yaitu det A ≠ 0. Sebaliknya, jika A matriks singular (det A = 0) maka matriks ini tidak memiliki invers.
Misalkan matriks A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%20a%20&%20b%5C%5C%20c%20&%20d%20%5Cend%7Bbmatrix%7D" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." /> dan matriks B = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%20p%20&%20q%5C%5C%20r%20&%20s%20%5Cend%7Bbmatrix%7D" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." /> sehingga berlaku A × B = B × A = I. Kita akan mencari elemen-elemen matriks B, yaitu p, q, r, dan s.
Dari persamaan A × B = I, diperoleh :
Jadi, diperoleh sistem persamaan :
ap + br = 1 dan aq + bs = 0
cp + dr = 0 cq + ds = 1
Dengan menyelesaikan sistem persamaan tersebut, kalian peroleh :
Dengan demikian,
Matriks B memenuhi A × B = I.
Sekarang, akan kita buktikan apakah matriks B × A = I?
Karena ad – bc ≠ 0, berlaku B × A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%201%20&%200%5C%5C%200%20&%201%20%5Cend%7Bbmatrix%7D" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." /> = I
Karena A × B = B × A = I maka B = A–1.
Jadi, jika A =
untuk ad – bc ≠ 0.
Contoh Soal 18 :
Tentukan invers matriks-matriks berikut.
Tentukan invers matriks-matriks berikut.
a. A =
b. B = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%203%20&%20-2%5C%5C%205%20&%20-4%20%5Cend%7Bbmatrix%7D" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." />
Jawaban :
Menentukan Invers Matriks Berordo 3x3 (Pengayaan)
Invers matriks berordo 3 × 3 dapat dicari dengan beberapa cara. Pada pembahasan kali ini kita akan menggunakan cara adjoin dan transformasi baris elementer.
a. Dengan Adjoin
Pada subbab sebelumnya, telah dijelaskan mengenai determinan matriks. Selanjutnya, adjoin A dinotasikan adj (A), yaitu transpose dari matriks yang elemen-elemennya merupakan kofaktor-kofaktor dari elemen-elemen matriks A, yaitu :
adj(A) = (kof(A))T
Adjoin A dirumuskan sebagai berikut.
Invers matriks persegi berordo 3 × 3 dirumuskan sebagai berikut.
Adapun bukti tentang rumus ini akan kalian pelajari lebih mendalam dijenjang pendidikan yang lebih tinggi.
Contoh Soal 19 :
Diketahui matriks A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%201%20&%202&1%5C%5C%202%20&%203&4%5C%5C%201%20&%202&3%20%5Cend%7Bbmatrix%7D" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." /> . Tentukan invers matriks A, misalnya kita gunakan perhitungan menurut baris pertama.
Jawaban :
Terlebih dahulu kita hitung determinan A.
det A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20det%5C:%20A=1%5Cbegin%7Bvmatrix%7D%203%20&%204%5C%5C%202%20&%203%20%5Cend%7Bvmatrix%7D-2%5Cbegin%7Bvmatrix%7D%202%20&%204%5C%5C%201%20&%203%20%5Cend%7Bvmatrix%7D+1%5Cbegin%7Bvmatrix%7D%202%20&%203%5C%5C%201%20&%202%20%5Cend%7Bvmatrix%7D" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." />
= 1(1) – 2(2) + 1(1) = –2
Dengan menggunakan rumus adjoin A, diperoleh :
adj(A) = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%201%20&%20-4&%205%5C%5C%20-2%20&%202&%20-2%5C%5C%201%20&%200&%20-1%20%5Cend%7Bbmatrix%7D" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." />
Jadi, A–1 dapat dihitung sebagai berikut.
b. Dengan Transformasi Baris Elementer
Untuk menentukan invers matriks An dengan cara transformasi baris elementer, dapat dilakukan dengan langkah-langkah berikut berikut.
1) Bentuklah matriks (An | In), dengan In adalah matriks identitas ordo n.
2) Transformasikan matriks (An | In) ke bentuk (In | Bn), dengan transformasi elemen baris.
3) Hasil dari Langkah 2, diperoleh invers matriks An adalah Bn.
Notasi yang sering digunakan dalam transformasi baris elementer adalah :
a) Bi ↔ Bj : menukar elemen-elemen baris ke-i dengan elemen-elemen baris ke-j;
b) k.Bi : mengalikan elemen-elemen baris ke-i dengan skalar k;
c) Bi + kBj : jumlahkan elemen-elemen baris ke-i dengan k kali elemen-elemen baris ke-j.
Contoh Soal 20 :
Tentukan invers matriks A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%202%20&%201%5C%5C%205%20&%203%20%5Cend%7Bbmatrix%7D" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." /> dengan transformasi baris elementer.
Penyelesaian :
Jadi, diperoleh A–1 = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%203%20&%20-1%5C%5C%20-5%20&%202%20%5Cend%7Bbmatrix%7D" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." />
Keterangan :
1/2 B1 : Kalikan elemen-elemen baris ke-1 dengan 1/2.
B2 – 5B1 : Kurangkan baris ke-2 dengan 5 kali elemen-elemen baris ke-1.
B1 – B2 : Kurangi elemen-elemen baris ke-1 dengan elemen-elemen baris ke-2.
2B2 : Kalikan elemen-elemen baris ke-2 dengan 2.
Contoh Soal 21 :
Tentukan invers matriks A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%201%20&%201&%200%5C%5C%202%20&%203&%202%5C%5C%202%20&%201&%203%20%5Cend%7Bbmatrix%7D" style="font-family:Arial,Helvetica,sans-serif; font-size:12px; height:auto; margin:0px; max-width:100%; text-align:center; vertical-align:middle" title="This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program." /> dengan transformasi baris elementer.
Jawaban :