Soal-Soal Matematika/Matriks

Bentuk dan sifat matriks

bentuk

• ordo 2x2: ${\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}$

• ordo 3x3: ${\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\\\end{bmatrix}}$

sifat

1. komutatif

• A + B = B + A

1. asosiatif

• (A + B) + C = A + (B + C)
• (A . B) . C = A. (B x C)

1. distributif

• A . (B + C) = A . B + A . C
• A . (B - C) = A . B - A . C

1. (k . A) . B = k. (A . B)
2. A . B ≠ B . A
3. A . I = A
4. A . A^-1 = I
5. (A . B)^-1 = B^-1 . A^-1
6. (A . B . C)^-1 = C^-1 . B^-1 . A^-1
7. det (A^-1) = det A (invers bukan pangkat)
8. det (A^T) = 1/det A
9. det (Aⁿ) = (det A)ⁿ

vektor baris

${\begin{bmatrix}3&7&2\end{bmatrix}}$

vektor kolom

${\begin{bmatrix}4\\1\\8\\\end{bmatrix}}$

matriks persegi

${\begin{bmatrix}9&13&5\\1&11&7\\2&6&3\\\end{bmatrix}}$

${\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\\\end{bmatrix}}$

• baris:

1. pertama: a11, a12 dan a13
2. kedua: a21, a22 dan a23
3. ketiga: a31, a32 dan a33

• kolom:

1. pertama: a11, a21 dan a31
2. kedua: a12, a22 dan a32
3. ketiga: a13, a23 dan a33

• diagonal

1. sisi kiri ke kanan: a11, a22 dan a33
2. sisi kanan ke kiri: a13, a22 dan a31

• Matriks diagonal

${\begin{bmatrix}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\\\end{bmatrix}}$

• Matriks segitiga bawah

${\begin{bmatrix}a_{11}&0&0\\a_{21}&a_{22}&0\\a_{31}&a_{32}&a_{33}\\\end{bmatrix}}$

• Matriks segitiga atas

${\begin{bmatrix}a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&0&a_{33}\\\end{bmatrix}}$

matriks perkalian

${\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}\cdot {\begin{bmatrix}e&f\\g&h\\\end{bmatrix}}$ = ${\begin{bmatrix}ae+bg&af+bh\\ce+dg&cf+dh\\\end{bmatrix}}$

ordo 2x2

bentuk

${\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}$

Matriks transpos (A^T)

${\begin{bmatrix}a&c\\b&d\end{bmatrix}}$

Determinan (Det)

ad - bc

Matriks singular adalah matriks yang hasil determinan bernilai nol sedangkan matriks nonsingular adalah matriks yang hasil determinan bernilai bukan nol.

Matriks inverse (A^-1)

${\frac {1}{\text{det A}}}$${\begin{bmatrix}d&-b\\-c&a\\\end{bmatrix}}$

ordo 3x3

bentuk

${\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\\\end{bmatrix}}$

Matriks transpos (A^T)

${\begin{bmatrix}a&d&g\\b&e&h\\c&f&i\\\end{bmatrix}}$

Determinan (Det)

• dengan sarrus

${\begin{aligned}{\begin{array}{rrr|rr}a&b&c&a&b\\d&e&f&d&e\\g&h&i&g&h\\-&-&-+&+&+\\\end{array}}\\\end{aligned}}$

det A = aei + bfg + cdh - bdi - afh - ceg

• dengan minor-kofaktor

untuk minor M_ij = det A_ij

untuk kofaktor C_ij = (-1)^i+j . M_ij

det A = $\sum _{j=1}^{n}i=a_{ij}\cdot C_{ij}$ dimana sembarang baris i atau kolom j (i atau j = 1, 2, 3, ..., n)

Adjoint (Adj)

kof (A) = (-1)^i+j . M_ij

kof (A) = ${\begin{bmatrix}M_{11}&M_{12}&M_{13}\\M_{21}&M_{22}&M_{23}\\M_{31}&M_{32}&M_{33}\\\end{bmatrix}}$

adj A = (kof (A))^T

Matriks inverse (A^-1)

• dengan adjoint

${\frac {adjA}{\text{det A}}}$

• dengan elementer

A | I diubah menjadi I | A^-1

contoh

1. tentukan hasil determinan serta matriks invers dari ${\begin{bmatrix}2&1\\6&4\\\end{bmatrix}}$!

Jawaban

$${\begin{aligned}*detA&=2\cdot 4-1\cdot 6=8-6=2\\*A^{-1}&={\frac {1}{detA}}\cdot {\begin{bmatrix}4&-1\\-6&2\\\end{bmatrix}}\\&={\frac {1}{2}}\cdot {\begin{bmatrix}4&-1\\-6&2\\\end{bmatrix}}\\&={\begin{bmatrix}2&-{\frac {1}{2}}\\-3&1\\\end{bmatrix}}\\\end{aligned}}$$

1. tentukan hasil determinan serta matriks invers dari ${\begin{bmatrix}1&2&3\\2&8&7\\1&5&6\\\end{bmatrix}}$!

Jawaban

$${\begin{aligned}*{\text{cara 1}}\\{\begin{array}{rrr|rr}1&2&3&1&2\\2&8&7&2&8\\1&5&6&1&5\\-&-&-+&+&+\\\end{array}}\\detA&=1\cdot 8\cdot 6+2\cdot 7\cdot 1+3\cdot 2\cdot 5-2\cdot 2\cdot 6-1\cdot 7\cdot 5-3\cdot 8\cdot 1=48+14+30-24-35-24=9\\{\text{cara 2}}\\detA&=a_{11}\cdot c_{11}+a_{12}\cdot c_{12}+a_{13}\cdot c_{13}\\&=1\cdot (-1)^{1+1}\cdot {\begin{bmatrix}8&7\\5&6\\\end{bmatrix}}+2\cdot (-1)^{1+2}\cdot {\begin{bmatrix}2&7\\1&6\\\end{bmatrix}}+3\cdot (-1)^{1+3}\cdot {\begin{bmatrix}2&8\\1&5\\\end{bmatrix}}\\&=1\cdot 1\cdot (48-35)+2\cdot (-1)\cdot (12-7)+3\cdot 1\cdot (10-8)\\&=13-10+6=9\\*{\text{cara 1}}\\kof(A)&={\begin{bmatrix}{\begin{bmatrix}8&7\\5&6\\\end{bmatrix}}&-{\begin{bmatrix}2&7\\1&6\\\end{bmatrix}}&{\begin{bmatrix}2&8\\1&5\\\end{bmatrix}}\\-{\begin{bmatrix}2&3\\5&6\\\end{bmatrix}}&{\begin{bmatrix}1&3\\1&6\\\end{bmatrix}}&-{\begin{bmatrix}1&2\\1&5\\\end{bmatrix}}\\{\begin{bmatrix}2&3\\8&7\\\end{bmatrix}}&-{\begin{bmatrix}1&3\\2&7\\\end{bmatrix}}&{\begin{bmatrix}1&2\\2&8\\\end{bmatrix}}\\\end{bmatrix}}\\&={\begin{bmatrix}13&-5&2\\3&3&-3\\-10&-1&4\\\end{bmatrix}}\\adj(A)&=(kof(A))^{T}\\&={\begin{bmatrix}13&3&-10\\-5&3&-1\\2&-3&4\\\end{bmatrix}}\\A^{-1}&={\frac {adjA}{detA}}\\&={\frac {1}{9}}\cdot {\begin{bmatrix}13&3&-10\\-5&3&-1\\2&-3&4\\\end{bmatrix}}\\&={\begin{bmatrix}{\frac {13}{9}}&{\frac {1}{3}}&-{\frac {10}{9}}\\-{\frac {5}{9}}&{\frac {1}{3}}&-{\frac {1}{9}}\\{\frac {2}{9}}&-{\frac {1}{3}}&{\frac {4}{9}}\\\end{bmatrix}}\\{\text{cara 2}}\\{\begin{array}{rrr|rrr}1&2&3&1&0&0\\2&8&7&0&1&0\\1&5&6&0&0&1\\\end{array}}\\{\text{b2-2b1}}\\{\begin{array}{rrr|rrr}1&2&3&1&0&0\\0&4&1&-2&1&0\\1&5&6&0&0&1\\\end{array}}\\{\text{b3-b1}}\\{\begin{array}{rrr|rrr}1&2&3&1&0&0\\0&4&1&-2&1&0\\0&3&3&-1&0&1\\\end{array}}\\{\text{1/4b2}}\\{\begin{array}{rrr|rrr}1&2&3&1&0&0\\0&1&{\frac {1}{4}}&-{\frac {1}{2}}&{\frac {1}{4}}&0\\0&3&3&-1&0&1\\\end{array}}\\{\text{b1-2b2}}\\{\begin{array}{rrr|rrr}1&0&{\frac {5}{2}}&2&-{\frac {1}{2}}&0\\0&1&{\frac {1}{4}}&-{\frac {1}{2}}&{\frac {1}{4}}&0\\0&3&3&-1&0&1\\\end{array}}\\{\text{b3-3b2}}\\{\begin{array}{rrr|rrr}1&0&{\frac {5}{2}}&2&-{\frac {1}{2}}&0\\0&1&{\frac {1}{4}}&-{\frac {1}{2}}&{\frac {1}{4}}&0\\0&0&{\frac {9}{4}}&{\frac {1}{2}}&-{\frac {3}{4}}&1\\\end{array}}\\{\text{4/9b3}}\\{\begin{array}{rrr|rrr}1&0&{\frac {5}{2}}&2&-{\frac {1}{2}}&0\\0&1&{\frac {1}{4}}&-{\frac {1}{2}}&{\frac {1}{4}}&0\\0&0&1&{\frac {2}{9}}&-{\frac {1}{3}}&{\frac {4}{9}}\\\end{array}}\\{\text{b1-5/2b3}}\\{\begin{array}{rrr|rrr}1&0&0&{\frac {13}{9}}&{\frac {1}{3}}&-{\frac {10}{9}}\\0&1&{\frac {1}{4}}&-{\frac {1}{2}}&{\frac {1}{4}}&0\\0&0&1&{\frac {2}{9}}&-{\frac {1}{3}}&{\frac {4}{9}}\\\end{array}}\\{\text{b2-1/4b3}}\\{\begin{array}{rrr|rrr}1&0&0&{\frac {13}{9}}&{\frac {1}{3}}&-{\frac {10}{9}}\\0&1&0&-{\frac {5}{9}}&{\frac {1}{3}}&-{\frac {1}{9}}\\0&0&1&{\frac {2}{9}}&-{\frac {1}{3}}&{\frac {4}{9}}\\\end{array}}\\{\text{jadi}}A^{-1}&={\begin{bmatrix}{\frac {13}{9}}&{\frac {1}{3}}&-{\frac {10}{9}}\\-{\frac {5}{9}}&{\frac {1}{3}}&-{\frac {1}{9}}\\{\frac {2}{9}}&-{\frac {1}{3}}&{\frac {4}{9}}\\\end{bmatrix}}\\\end{aligned}}$$