Rumus Matriks – Perkalian, Penjumlahan, Pengurangan – Operasi Perhitungan Matriks – Contoh Soal dan Jawaban

Pinter Pandai

6 tahun ago

Matriks

Adalah sekumpulan bilangan yang disusun secara baris dan kolom dan ditempatkan pada kurung biasa atau kurung siku. Dibawah ini Anda dapat menemukan operasi perhitungan matriks, beserta contoh soal dan jaabannya.

Penulisan matriks:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04c3b89a7b1136bab7b89428e485cfcc97378be1" alt="{\displaystyle {\begin{pmatrix}2&3\\1&4\end{pmatrix}}}" width="3797.5" height="2659.1">$

atau

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7808a7b22fc594f00d00ba5e3849b799b88ee3" alt="{\displaystyle {\begin{bmatrix}2&3\\1&4\end{bmatrix}}}" width="3381.5" height="2659.1">$

Ordo suatu matriks adalah bilangan yang menunjukkan banyaknya baris (m) dan banyaknya kolom (n).

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a7348fa51e8510942e776d7a32b527fe12a2f1" alt="{\displaystyle {\begin{pmatrix}2&3&5\\1&4&-7\end{pmatrix}}}" width="6076.5" height="2659.1">$ Matriks di atas berordo 3×2.

Matriks Identitas (I)

Matriks identitas (I)adalah matriks yang nilai-nilai elemen pada diagonal utama selalu 1.

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deedb7d1ff5ce518d7d26f1f3a376dc3457d6ce3" alt="{\displaystyle I={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}" width="7414.6" height="3950.7">$

Matriks Transpose (A^t)

Matriks transpose adalah matriks yang mengalami pertukaran elemen dari baris menjadi kolom dan sebaliknya. Contoh:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3445650d735de7344d722745b624ae41a16b692d" alt="{\displaystyle A={\begin{pmatrix}2&3&5\\1&4&-7\end{pmatrix}}}" width="8161.1" height="2659.1">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be675e3c58ad35dce2c6db55f1da2fe6ca02048" alt="{\displaystyle A^{t}={\begin{pmatrix}2&1\\3&4\\5&-7\end{pmatrix}}}" width="7294.2" height="3950.7">$

Operasi Perhitungan Matriks

Kesamaan 2 matriks

2 matriks dikatakan sama jika ordonya sama dan elemen yang seletak sama.

Contoh: $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b038475838e11e017343696b85c53f9ba9e60201" alt="{\displaystyle {\begin{pmatrix}2&3&5\\1&4&-7\end{pmatrix}}={\begin{pmatrix}2&6x&z-y\\2y+2&4&-7\end{pmatrix}}}" width="17190.5" height="2659.1">$

Tentukan nilai 2x-y+5z!

Jawab:

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2823014b08c738d0755d2397dc6d997a7e59c6c4" alt="{\displaystyle x={\frac {1}{2}}}" width="2767.1" height="2228.5">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/974352c87d2118c4894857fe9cfcdf3db664c809" alt="{\displaystyle y=-{\frac {1}{2}}}" width="3470.6" height="2228.5">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2f29fb3f1e2fd416516d2dc25fee570f93adaf" alt="{\displaystyle z={\frac {9}{2}}}" width="2663.1" height="2228.5">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/590ac40ce1eef53d48e19ffe55c97bb4a98a78db" alt="{\displaystyle 2x-y+5z}" width="4985.4" height="1080.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/145aa8ee836e4bdf10a93a541d2d460d9a0984f3" alt="{\displaystyle =2\left({\frac {1}{2}}\right)-{\frac {1}{2}}+5\left({\frac {9}{2}}\right)}" width="10364" height="2659.1">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e78f529856bcc6db8784c5c7ce9dd78797ea67" alt="{\displaystyle =23}" width="2057.3" height="936.9">

Penjumlahan matriks

2 matriks bisa dijumlahkan jika ordonya sama dan penjumlahan dilakukan dengan cara menjumlahkan elemen yang seletak.

Contoh: $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1f34298c41a965d5cb56474401b23c818ea33a2" alt="{\displaystyle {\begin{pmatrix}2&3&5\\1&4&-7\end{pmatrix}}+{\begin{pmatrix}2&6x&z-y\\2y+2&4&-7\end{pmatrix}}={\begin{pmatrix}4&3+6x&5+z-y\\2y+3&8&-14\end{pmatrix}}}" width="31640.3" height="2659.1">$

Pengurangan matriks

2 matriks bisa dikurangkan jika ordonya sama dan pengurangan dilakukan dengan cara mengurangkan dari elemen yang seletak.

Contoh: $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7bce9486f0dc9c2b7c17da590c46e8b4d4bf3fc" alt="{\displaystyle {\begin{pmatrix}2&3&5\\1&4&-7\end{pmatrix}}-{\begin{pmatrix}2&6x&z-y\\2y+2&4&-7\end{pmatrix}}={\begin{pmatrix}0&3-6x&5-z-y\\-2y-1&0&0\end{pmatrix}}}" width="32418.8" height="2659.1">$

Perkalian bilangan dengan matriks

Contoh:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122b7938f48604b90e67835390bbc4f7ebdb12d4" alt="{\displaystyle 3{\begin{pmatrix}2&6x&z-y\\2y+2&4&-7\end{pmatrix}}={\begin{pmatrix}6&18x&3z-3y\\6y+6&12&-21\end{pmatrix}}}" width="22895.9" height="2659.1">$

Perkalian matriks

2 Matriks dapat dikalikan jika jumlah baris matriks A = jumlah kolom matriks B.

Definisi Perkalian Matriks

Jika $A$ adalah matriks $n \times m$ dan $B$ adalah matriks $m \times p$ ,

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b1644351bc2041175b19cbc65da03ef78130c7" alt="{\displaystyle \mathbf {A} ={\begin{pmatrix}a_{11}&amp;a_{12}&amp;\cdots &amp;a_{1m}\\a_{21}&amp;a_{22}&amp;\cdots &amp;a_{2m}\\\vdots &amp;\vdots &amp;\ddots &amp;\vdots \\a_{n1}&amp;a_{n2}&amp;\cdots &amp;a_{nm}\\\end{pmatrix}},\quad \mathbf {B} ={\begin{pmatrix}b_{11}&amp;b_{12}&amp;\cdots &amp;b_{1p}\\b_{21}&amp;b_{22}&amp;\cdots &amp;b_{2p}\\\vdots &amp;\vdots &amp;\ddots &amp;\vdots \\b_{m1}&amp;b_{m2}&amp;\cdots &amp;b_{mp}\\\end{pmatrix}}}" width="27522.7" height="6390.6">

Produk matriks $C = AB$ adalah matriks $n \times p$ .

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ac0c831c365b7424cc43239aae8cebea27c56c" alt="{\displaystyle \mathbf {C} ={\begin{pmatrix}c_{11}&amp;c_{12}&amp;\cdots &amp;c_{1p}\\c_{21}&amp;c_{22}&amp;\cdots &amp;c_{2p}\\\vdots &amp;\vdots &amp;\ddots &amp;\vdots \\c_{n1}&amp;c_{n2}&amp;\cdots &amp;c_{np}\\\end{pmatrix}}}" width="12460.8" height="6390.6">

sehingga

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cfeccef1c8c7e6da0ddf08daed8dbf3c6f50c5e" alt="{\displaystyle c_{ij}=a_{i1}b_{1j}+\cdots +a_{im}b_{mj}=\sum _{k=1}^{m}a_{ik}b_{kj},}" width="17018.3" height="2946.1">

for $i = 1, \dots, n$ dan $j = 1, \dots, p$ .

2 Matriks dapat dikalikan jika jumlah baris matriks A = jumlah kolom matriks B.
Penghitungan perkalian matriks:

Misalkan:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/493f5ae78f7a6c90982d33f44edcf74679abbd00" alt="{\displaystyle A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}}" width="5934.1" height="2659.1">$ dan $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dca2cc27b098220d979999e458ccefa747905e25" alt="{\displaystyle B={\begin{pmatrix}p&q\\r&s\end{pmatrix}}}" width="5863.1" height="2659.1">$

maka $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a29a4e47ab1f554258bed3ba8a13803140d9c517" alt="{\displaystyle A\times B={\begin{pmatrix}ap+br&aq+bs\\cp+dr&cq+ds\end{pmatrix}}}" width="13112.4" height="2659.1">$

Contoh:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf541a705c57b4d40981219634df8658dadc7c8" alt="{\displaystyle {\begin{pmatrix}2&6\\3&4\end{pmatrix}}{\begin{pmatrix}9&8\\2&10\end{pmatrix}}={\begin{pmatrix}30&76\\35&64\end{pmatrix}}}" width="14228.2" height="2659.1">$

Pembagian matriks

Sebenarnya kita tidak benar-benar membagi matriks, kita melakukannya dengan cara ini:

A/B = A × (1/B) = A × B^-1

yang dimana B^-1 berarti the “kebalikan” dari B.

Jadi kita tidak “membagi” dalam perhitungan matriks, malah kita kalikan dengan invers. Dan ada cara khusus untuk menemukan Invers yang dapat Anda temukan di baah ini.

Determinan suatu matriks

Matriks ordo 2×2

Misalkan:

maka Determinan A (ditulis $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e741013c6ab037c83c15fd331d421ff0e455d2e" alt="{\displaystyle \left\vert A\right\vert }" width="1307.5" height="1223.9">$ ) adalah:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb5c4549ad44e8f44b06890f33ad7fe717579b7" alt="{\displaystyle \left\vert A\right\vert =a\times d-b\times c}" width="8226.4" height="1223.9">$

Matriks ordo 3×3

Cara Sarrus

Misalkan:

Jika $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a1a1eace3a4f1ff180a1cf00e73cd8965a0ebf7" alt="{\displaystyle A={\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}}}" width="7815.6" height="4094.3">$ maka tentukan $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e741013c6ab037c83c15fd331d421ff0e455d2e" alt="{\displaystyle \left\vert A\right\vert }" width="1307.5" height="1223.9">$ !

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d8aac083a151d5f0a02851a813f4d1ea04eb95" alt="{\displaystyle \left\vert A\right\vert =\left\vert {\begin{matrix}a&b&c\\d&e&f\\g&h&i\end{matrix}}\right\vert {\begin{matrix}a&b\\d&e\\g&h\end{matrix}}}" width="9774.8" height="4094.3">$

Penghitungan matriks dilakukan dengan cara menambahkan elemen dari kiri atas ke kanan bawah (mulai dari a → e → i, b → f → g, dan c → d → h) lalu dikurangi dengan elemen dari kanan atas ke kiri bawah (mulai dari c → e → g, a → f → h, dan b → d → i) sehingga menjadi:

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1593095d98c9f645fc41283ed666571b9e53aff" alt="{\displaystyle \left\vert A\right\vert =a.e.i+b.f.g+c.d.h-g.e.c-h.f.a-i.d.b}" width="22769.3" height="1223.9">

Contoh:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88989e78850e80b1c39a273832644aab0abf69a6" alt="{\displaystyle A={\begin{pmatrix}-2&0&1\\3&2&-1\\1&-3&5\end{pmatrix}}}" width="9996.1" height="3950.7">$ maka tentukan $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e741013c6ab037c83c15fd331d421ff0e455d2e" alt="{\displaystyle \left\vert A\right\vert }" width="1307.5" height="1223.9">$ !

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/478417ae5c457c896c7e31c8656b6e581f3093e1" alt="{\displaystyle \left\vert A\right\vert =\left\vert {\begin{matrix}-2&0&1\\3&2&-1\\1&-3&5\end{matrix}}\right\vert {\begin{matrix}-2&0\\3&2\\1&-3\end{matrix}}}" width="13407.3" height="3950.7">$

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1388ba640878546c8bb40d15f6efd1a95a6e5bc4" alt="{\displaystyle \left\vert A\right\vert =(-2.2.5)+(0.-1.-1)+(1.3.-3)-(1.2.1)-(-2.-1.-3)-(0.3.5)=-20+0-9-2+6-0=-25}" width="48297.3" height="1223.9">

Cara ekspansi baris-kolom

Misalkan:

Jika $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5264b249bd1af796c1c27f321353af9d8f85bd" alt="{\displaystyle P={\begin{pmatrix}-2&0&1\\3&2&-1\\1&-3&5\end{pmatrix}}}" width="9997.1" height="3950.7">$ maka tentukan $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b72f194696e139e7e0e04879edc07d8915aa97b1" alt="{\displaystyle \left\vert P\right\vert }" width="1308.5" height="1223.9">$ dengan ekspansi baris pertama!

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7aadd8c1e6966db3cd5911c40d9a707b7d33e84" alt="{\displaystyle \left\vert P\right\vert =-2\left\vert {\begin{matrix}2&-1\\-3&5\end{matrix}}\right\vert -0\left\vert {\begin{matrix}3&-1\\1&5\end{matrix}}\right\vert +1\left\vert {\begin{matrix}3&2\\1&-3\end{matrix}}\right\vert }" width="19627.1" height="2659.1">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b49a125798da89a4ab9bc1a860140dd5d6353c" alt="{\displaystyle \left\vert P\right\vert =-2(10-3)-0+1(-9-2)=-25}" width="17766.9" height="1223.9">$

Matriks Singular

Matriks singular adalah matriks yang nilai determinannya 0.

Contoh:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ee1c8f32832d50c1259e6675d13693086f42ad" alt="{\displaystyle P={\begin{pmatrix}-4&5x\\-x&20\end{pmatrix}}}" width="7306.1" height="2659.1">$

Jika A matriks singular, tentukan nilai x!

Jawab:

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e05b0eb12b39628c5edeaa231d0667014c57dbe3" alt="{\displaystyle -80+5x^{2}=0}" width="6363.9" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07302c0557ff7ef673337825a38ae6c8a005004" alt="{\displaystyle 5(x^{2}-16)=0}" width="6364.4" height="1367.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8431bbc5ccea24d6596545276a84cca29b01d2d9" alt="{\displaystyle x=-4}" width="3185.6" height="1008.6">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73bfdd8100b6a4ce07d011900560f102e3965064" alt="{\displaystyle x=4}" width="2407.1" height="936.9">

Invers Matriks

Invers matriks 2×2

Misalkan:

maka inversnya adalah:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10147164b4fe39cf848c5d7b70f5d5e47afae411" alt="{\displaystyle A^{-1}={\frac {1}{\left\vert A\right\vert }}{\begin{pmatrix}d&-b\\-c&a\end{pmatrix}}={\frac {1}{a.d-b.c}}{\begin{pmatrix}d&-b\\-c&a\end{pmatrix}}}" width="20912.9" height="2730.8">$