Determinan Matriks

Syarat suatu matriks dapat dicari determinannya adalah matriks tersebut harus merupakan matriks persegi

a. Determinan Matriks Ordo 2 × 2

Misalkan 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." /> adalah matriks yang berordo 2 × 2 dengan elemen a dan d terletak pada diagonal utama pertama, sedangkan b dan c terletak pada diagonal kedua. Determinan matriks A dinotasikan ”det A” atau |A| adalah suatu bilangan yang diperoleh dengan mengurangi hasil kali elemen-elemen pada diagonal utama dengan hasil kali elemen-elemen diagonal kedua.

Dengan demikian, dapat diperoleh rumus det A sebagai berikut.

det A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bvmatrix%7D%20a%20&b%20%5C%5C%20c%20&%20d%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." /> = ad – bc

Contoh Soal 1 :

Tentukan determinan matriks-matriks berikut.

a. A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%205%20&%202%5C%5C%204%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." /> b. B = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%20-4%20&%20-1%5C%5C%203%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." />

Penyelesaian :

a. det A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%205%20&%202%5C%5C%204%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." /> = (5 × 3) – (2 × 4) = 7

b. det B = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%20-4%20&%20-1%5C%5C%203%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." /> = ((–4) × 2) – (3 × (–1)) = – 5

b. Determinan Matriks Ordo 3 × 3 (Pengayaan)

Jika A = alt="Matriks Ordo 3 × 3" id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%20a_%7B11%7D%20&%20a_%7B12%7D&a_%7B13%7D%5C%5C%20a_%7B21%7D%20&%20a_%7B22%7D&a_%7B23%7D%5C%5C%20a_%7B31%7D%20&%20a_%7B32%7D&a_%7B33%7D%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." /> adalah matriks persegi berordo 3 × 3, determinan A dinyatakan dengan det A = alt="Matriks Ordo 3 × 3" id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%20a_%7B11%7D%20&%20a_%7B12%7D&a_%7B13%7D%5C%5C%20a_%7B21%7D%20&%20a_%7B22%7D&a_%7B23%7D%5C%5C%20a_%7B31%7D%20&%20a_%7B32%7D&a_%7B33%7D%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." />

Ada 2 cara yang dapat digunakan untuk menentukan determinan matriks berordo 3 × 3, yaitu aturan Sarrus dan metode minor-kofaktor.

Aturan Sarrus

Untuk menentukan determinan dengan aturan Sarrus, perhatikan alur berikut. Misalnya, kita akan menghitung determinan matriks A_{3 × 3}. Gambaran perhitungannya adalah sebagai berikut.

alt="" src="http://www.utakatikotak.com/public_assets/upload/images/20170119/Aturan-Sarrus-1652013.jpg" style="height:177px; width:400px" />

Metode Minor-Kofaktor

Misalkan matriks A dituliskan dengan [a_ij]. Minor elemen aij yang dinotasikan dengan M_ij adalah determinan setelah elemen-elemen baris ke-i dan kolom ke-j dihilangkan. Misalnya, dari matriks A_{3 × 3} kita hilangkan baris ke-2 kolom ke-1 sehingga :

alt="" src="http://www.utakatikotak.com/public_assets/upload/images/20170119/Metode-Minor-Kofaktor-matriks-1652013.jpg" style="height:186px; width:320px" />

Akan diperoleh M₂₁ = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%20a_%7B12%7D&a_%7B13%7D%5C%5C%20a_%7B32%7D&a_%7B33%7D%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." /> . M₂₁ adalah minor dari elemen matriks A baris ke-2 kolom ke-1 atau M₂₁ = minor a₂₁. Sejalan dengan itu, kita dapat memperoleh minor yang lain, misalnya :

M₁₃ = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%20a_%7B21%7D&a_%7B12%7D%5C%5C%20a_%7B31%7D&a_%7B32%7D%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." />

Kofaktor elemen aij, dinotasikan K_ij adalah hasil kali (–1)^i+j dengan minor elemen tersebut. Dengan demikian, kofaktor suatu matriks dirumuskan dengan :

K_ij = (–1)^i+j M_ij

Dari matriks A di atas, kita peroleh misalnya kofaktor a₂₁ dan a₁₃berturut-turut adalah

K₂₁ = (–1)²⁺¹ M₂₁ = –M₂₁ = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%20a_%7B21%7D&a_%7B12%7D%5C%5C%20a_%7B31%7D&a_%7B32%7D%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." />

K₁₃ = (–1)¹⁺³ M₁₃ = M₁₃ = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%20a_%7B21%7D&a_%7B22%7D%5C%5C%20a_%7B31%7D&a_%7B32%7D%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." />

Kofaktor dari matriks A_{3 × 3} adalah kof(A) = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%20K_%7B11%7D%20&%20K_%7B12%7D&K_%7B13%7D%5C%5C%20K_%7B21%7D%20&%20K_%7B22%7D&K_%7B23%7D%5C%5C%20K_%7B31%7D%20&%20K_%7B32%7D&K_%7B33%7D%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." />

Nilai dari suatu determinan merupakan hasil penjumlahan dari perkalian elemen-elemen suatu baris (atau kolom) dengan kofaktornya. Untuk menghitung determinan, kita dapat memilih dahulu sebuah baris (atau kolom) kemudian kita gunakan aturan di atas. Perhatikan cara menentukan determinan berikut.

Misalkan diketahui matriks A = alt="Matriks Ordo 3 × 3" id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%20a_%7B11%7D%20&%20a_%7B12%7D&a_%7B13%7D%5C%5C%20a_%7B21%7D%20&%20a_%7B22%7D&a_%7B23%7D%5C%5C%20a_%7B31%7D%20&%20a_%7B32%7D&a_%7B33%7D%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." />

Determinan matriks A dapat dihitung dengan cara berikut.

Kita pilih baris pertama sehingga

det A = a₁₁ K₁₁ + a₁₂ K₁₂ + a₁₃ K₁₃

= a₁₁ (–1)¹⁺¹ M₁₁ + a₁₂ (–1)¹⁺² M₁₂ + a₁₃ (–1)¹⁺³ M₁₃
₌

id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20a_%7B11%7D%5Cbegin%7Bbmatrix%7D%20a_%7B22%7D%20&%20a_%7B23%7D%5C%5C%20a_%7B32%7D%20&%20a_%7B33%7D%20%5Cend%7Bbmatrix%7D-a_%7B12%7D%5Cbegin%7Bbmatrix%7D%20a_%7B21%7D%20&%20a_%7B23%7D%5C%5C%20a_%7B31%7D%20&%20a_%7B33%7D%20%5Cend%7Bbmatrix%7D+a_%7B13%7D%5Cbegin%7Bbmatrix%7D%20a_%7B21%7D%20&%20a_%7B22%7D%5C%5C%20a_%7B31%7D%20&%20a_%7B32%7D%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." />

= a₁₁(a₂₂ a₃₃ – a₃₂ a₂₃) – a₁₂(a₂₁ a₃₃ – a₃₁ a₂₃) + a₁₃(a₂₁ a₃₂ – a₃₁ a₂₂)

= a₁₁ a₂₂ a₃₃ – a₁₁ a₂₃ a₃₂ – a₁₂ a₂₁ a₃₃ + a₁₂ a₂₃ a₃₁ + a₁₃ a₂₁ a₃₂ – a₁₃ a₂₂ a₃₁

= a₁₁ a₂₂ a₃₃ + a₁₂ a₂₃ a₃₁ + a₁₃ a₂₁ a₃₂ – a₁₃ a₂₂ a₃₁ – a₁₁ a₂₃ a₃₂ – a₁₂ a₂₁ a₃₃

Tampak bahwa det A matriks ordo 3 × 3 yang diselesaikan dengan cara minor kofaktor hasilnya sama dengan det A menggunakan cara Sarrus.

Contoh Soal 2 :

Tentukan determinan dari matriks A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bvmatrix%7D%201%20&2%20&3%20%5C%5C%202%20&1%20&4%20%5C%5C%203&%201&%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." /> dengan aturan Sarrus dan minor-kofaktor.

Penyelesaian :

Cara 1: (Aturan Sarrus)

det A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bvmatrix%7D%201%20&2%20&3%20%5C%5C%202%20&1%20&4%20%5C%5C%203&%201&%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 × 4 × 3) + (3 × 2 × 1) – (3 × 1 × 3)

– (1 × 4 × 1) – (2 × 2 × 2)

= 2 + 24 + 6 – 9 – 4 – 8

= 11

Cara 2: (Minor-kofaktor)

Misalnya kita pilih perhitungan menurut baris pertama sehingga diperoleh :

det A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%201%5Cbegin%7Bvmatrix%7D%201%20&4%20%5C%5C%201%20&%202%20%5Cend%7Bvmatrix%7D-2%5Cbegin%7Bvmatrix%7D%202%20&4%20%5C%5C%203%20&%202%20%5Cend%7Bvmatrix%7D+3%5Cbegin%7Bvmatrix%7D%202%20&1%20%5C%5C%203%20&%201%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." />

= –2 – 2(–8) + 3(–1)

= –2 + 16 – 3 = 11

Coba kalian selidiki nilai determinan ini dengan cara lain. Apakah hasilnya sama?

c. Sifat-Sifat Determinan Matriks

Berikut disajikan beberapa sifat determinan matriks

1. Jika semua elemen dari salah satu baris/kolom sama dengan nol maka determinan matriks itu nol.

Misal : id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20A=%5Cbegin%7Bbmatrix%7D%200%20&%200%5C%5C%202%20&%203%20%5Cend%7Bbmatrix%7D%5Crightarrow%20%5Cleft%20%7C%20A%20%5Cright%20%7C=0;%5C:%20B=%5Cbegin%7Bbmatrix%7D%202%20&%203%20&%201%5C%5C%200%20&%200%20&%200%5C%5C%205%20&%204%20&%201%20%5Cend%7Bbmatrix%7D%5Crightarrow%20%5Cleft%20%7C%20B%20%5Cright%20%7C=0" 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." />

2. Jika semua elemen dari salah satu baris/kolom sama dengan elemen-elemen baris/kolom lain maka determinan matriks itu nol.

Misal B = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%204%20&%203%20&%202%5C%5C%205%20&%207%20&%208%5C%5C%204%20&%203%20&%202%20%5Cend%7Bbmatrix%7D%5Crightarrow%20%5Cleft%20%7C%20B%20%5Cright%20%7C=0" 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 elemen-elemen baris ke-1 dan ke-3 sama).

3. Jika elemen-elemen salah satu baris/kolom merupakan kelipatan dari elemen-elemen baris/kolom lain maka determinan matriks itu nol.

Misal A = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cbegin%7Bbmatrix%7D%201%20&%202%20&%203%5C%5C%205%20&%207%20&%200%5C%5C%202%20&%204%20&%206%20%5Cend%7Bbmatrix%7D%5Crightarrow%20%5Cleft%20%7C%20A%20%5Cright%20%7C=0" 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 elemen-elemen baris ke-3 sama dengan kelipatan elemen-elemen baris ke-1).

4. |AB| = |A| ×|B|

5. |AT| = |A|, untuk AT adalah transpose dari matriks A.

6. |A^–1| = id="equationview" src="http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%5Cfrac%7B1%7D%7B%5Cleft%20%7C%20A%20%5Cright%20%7C%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." /> , untuk A^–1 adalah invers dari matriks A. (Materi invers akan kalian pelajari pada subbab berikutnya).

7. |kA| = kn |A|, untuk A ordo n × n dan k suatu konstanta. Sifat-sifat di atas tidak dibuktikan di sini. Pembuktian sifat-sifat ini akan kalian pelajari di jenjang yang lebih tinggi.

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