Selidiki bahwa det(K^n)=(det K)^n untuk matriks :------a. A= 4 1 3 -2 dengan n=4 b. A= 2 -1 3 1 2 .... Question from @aldienrev

October 2018 1 8K Report

Selidiki bahwa det(K^n)=(det K)^n untuk matriks :
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a. A= 4 1
3 -2

dengan n=4

b. A= 2 -1 3
1 2 4
5 -3 6

dengan n=2

catatan : didefinisikan K^n = K x K^n-1

Takamori37 Mari selidiki ~

a.)
Dengan A⁴
Berdasarkan hasil berikut:
$$\begin{align}A^2&=\left[\begin{array}{ccc}4&1\\3&-2\end{array}\right]\left[\begin{array}{ccc}4&1\\3&-2\end{array}\right]&&=\left[\begin{array}{ccc}16+3&4-2\\12-6&3+4\end{array}\right]&&=\left[\begin{array}{ccc}19&2\\6&7\end{array}\right] \\ A^3&=\left[\begin{array}{ccc}19&2\\6&7\end{array}\right]\left[\begin{array}{ccc}4&1\\3&-2\end{array}\right]&&=\left[\begin{array}{ccc}76+6&19-4\\24+21&6-14\end{array}\right]&&=\left[\begin{array}{ccc}82&15\\45&-8\end{array}\right]\end{align}$

$$\begin{align}A^4&=\left[\begin{array}{ccc}82&15\\45&-8\end{array}\right]\left[\begin{array}{ccc}4&1\\3&-2\end{array}\right]&&=\left[\begin{array}{ccc}328+45&82-30\\180-24&45+16\end{array}\right] \\ &=\left[\begin{array}{ccc}373&52\\156&61\end{array}\right]\end{align}$

Dengan demikian, cek validasi:
$$\begin{align}\det(A^4)&=^?(\det(A))^4 \\ 373\times61-156\times52&=^?(4\times(-2)-3\times1)^4 \\ 22.753-8.112&=^?(-8-3)^4 \\ 14.641&=^?(-11)^4 \\ 14.641&=14.641\ \ \boxed{\text{Valid}}\end{align}$

b.)
$$\begin{align}A^2&=\left[\begin{array}{ccc}2&-1&3\\1&2&4\\5&-3&6\end{array}\right]\left[\begin{array}{ccc}2&-1&3\\1&2&4\\5&-3&6\end{array}\right] \\ &=\left[\begin{array}{ccc}4-1+15&-2-2-9&6-4+18\\2+2+20&-1+4-12&3+8+24\\10-3+30&-5-6-18&15-12+36\end{array}\right] \\ &=\left[\begin{array}{ccc}18&-13&20\\24&-9&35\\37&-29&39\end{array}\right]\end{align}$

Cek validasi determinan:
$$\begin{align}\det(A^2)&=-6.318-16.835-13.920-[-6.660-18.270-12.168] \\ &= -37.073-(-37.098) \\ &=-37.073+37.098 \\ &=25\end{align}$

Dengan:
$$\begin{align}(\det(A))^2&=(24-20-9-(30-24-6))^2 \\ &=(24-29-0)^2 \\ &=(-5)^2 \\ &=25\end{align}$

Karena sama, maka:
$25&=25\ \ \boxed{\text{Valid}}$

aldienrev waduh panjang banget yaa, oke deh saya pelajarin dulu makasih kak :D

Takamori37 Oke, sama-sama. Silakan dipahami ~