Tentukan vektor eigen dari matrik
A= 7 0 0
0 3 0
0 0 3
Jawab:
Penjelasan dengan langkah-langkah:
Pengertian Eigen vektor : yaitu vektor-vektor yang mana akan sekedar mengalami dilatasi apabila suatu transformasi linear di terapkan pada vektor tersebut
[tex]\displaystyle \bold{A}= \left[\begin{array}{ccc}7&0&0\\0&3&0\\0&0&3\end{array}\right][/tex]
1. Jawaban singkat nya:
Karena matriks A bersifat diagonal, maka A akan memiliki 3 eigenvalue (2 kembar + 1, 2 kembar karena ada 2 bilangan yang sama di A), dimana tiap eigenvalue = bilangan di matriks A, dan secara otomatis memiliki 3 eigenvektor.
Jika ketiga vektor basis 3 dimensi :
[tex]\bold{e_1} = \left[\begin{array}{ccc}1\\0\\0\end{array}\right] , \bold{e_2} = \left[\begin{array}{ccc}0\\1\\0\end{array}\right], \bold{e_3} = \left[\begin{array}{ccc}0\\0\\1\end{array}\right][/tex]
diterapkan transformasi linear A maka akan didapatkan:
[tex]\bold{A\cdot e_1} = \left[\begin{array}{ccc}7\\0\\0\end{array}\right] , \bold{A\cdot e_2} = \left[\begin{array}{ccc}0\\3\\0\end{array}\right], \bold{A\cdot e_3} = \left[\begin{array}{ccc}0\\0\\3\end{array}\right][/tex]
maka eigenvektor dari a adalah ketiga vektor basis 3 Dimensi
[tex]\text{Eigenvektornya :}\\\\\bold{v_1} = \left[\begin{array}{ccc}1\\0\\0\end{array}\right] , \bold{v_2} = \left[\begin{array}{ccc}0\\1\\0\end{array}\right], \bold{v_3} = \left[\begin{array}{ccc}0\\0\\1\end{array}\right][/tex]
[tex]\text{solusi lebih umum nya :}\\\bold{v_1} = \left[\begin{array}{ccc}x_1\\0\\0\end{array}\right],\bold{v_2} = \left[\begin{array}{ccc}0\\y_2\\0\end{array}\right],\bold{v_3} = \left[\begin{array}{ccc}0\\0\\z_3\end{array}\right][/tex]
2. Jawaban padatnya :
[tex]\displaystyle \det(\bold{A-\text{k}\cdot I})=\det(\bold{M})=\left|\begin{array}{ccc}7-k&0&0\\0&3-k&0\\0&0&3-k\end{array}\right| = 0\\\\(7-k)(3-k)^2=0\to k = \{3,7\}\\\\\bold{M_1} = \left[\begin{array}{ccc}7-k_1&0&0\\0&3-k_1&0\\0&0&3-k_1\end{array}\right] \\\\\bold{M_1} = \left[\begin{array}{ccc}7-7&0&0\\0&3-7&0\\0&0&3-7\end{array}\right] =\left[\begin{array}{ccc}0&0&0\\0&-4&0\\0&0&-4\end{array}\right] \\\\[/tex]
[tex]\bold{M_2} = \left[\begin{array}{ccc}7-k_2&0&0\\0&3-k_2&0\\0&0&3-k_2\end{array}\right] \\\\\bold{M_2} = \left[\begin{array}{ccc}7-3&0&0\\0&3-3&0\\0&0&3-3\end{array}\right] =\left[\begin{array}{ccc}4&0&0\\0&0&0\\0&0&0\end{array}\right] \\\\[/tex]
- Eigen vektor 1 :
[tex]\bold{M_1\cdot v_1} = \left[\begin{array}{ccc}0&0&0\\0&-4&0\\0&0&-4\end{array}\right] \left[\begin{array}{ccc}x_1\\y_1\\z_1\end{array}\right] =\left[\begin{array}{ccc}0\\0\\0\end{array}\right] \to \left[\begin{array}{ccc|c}0&0&0&0\\0&-4&0&0\\0&0&-4&0\end{array}\right] \\\\\left[\begin{array}{ccc|c}0&0&0&0\\0&-4&0&0\\0&0&-4&0\end{array}\right] \xrightarrow{1) -\frac{R_2}{4}\to R_2, \;2) -\frac{R_3}{4}\to R_3}\left[\begin{array}{ccc|c}0&0&0&0\\0&1&0&0\\0&0&1&0\end{array}\right] \\\\[/tex]
[tex]\left[\begin{array}{ccc|c}0&0&0&0\\0&1&0&0\\0&0&1&0\end{array}\right] \text{diubah menjadi persamaan :}\\\\1) 0\cdot x_1+0\cdot y_1+0\cdot z_1=0 \to 0=0, x_1 = x_1\\2) y_1=0\\3) z_1 = 0\\\\\text{Eigenvektornya :}\\\\ \Huge{\boxed{\boxed{\bold{v_1} = \left[\begin{array}{ccc}1\\0\\0\end{array}\right]}}[/tex]
[tex]\text{solusi lebih umum nya :}\\\Huge{\boxed{\boxed{\bold{v_1} = \left[\begin{array}{ccc}x_1\\0\\0\end{array}\right]}}[/tex]
- Eigen vektor 2 dan 3:
[tex]\bold{M_2\cdot v} = \left[\begin{array}{ccc}4&0&0\\0&0&0\\0&0&0\end{array}\right] \left[\begin{array}{ccc}x_1\\y_1\\z_1\end{array}\right] =\left[\begin{array}{ccc}0\\0\\0\end{array}\right] \to \left[\begin{array}{ccc|c}4&0&0&0\\0&0&0&0\\0&0&0&0\end{array}\right] \\\\\left[\begin{array}{ccc|c}4&0&0&0\\0&0&0&0\\0&0&0&0\end{array}\right]\xrightarrow{1)\frac{R_1}{4}\to R_1}\left[\begin{array}{ccc|c}1&0&0&0\\0&0&0&0\\0&0&0&0\end{array}\right] \\\\[/tex]
[tex]\left[\begin{array}{ccc|c}1&0&0&0\\0&0&0&0\\0&0&0&0\end{array}\right] \text{diubah menjadi persamaan :}\\\\1) x_1=0\\2) \; \&\; 3) y_1=y_1, \;z_1 = z_1\\\\\text{Eigenvektornya :}\\\\ \Huge{\boxed{\boxed{\bold{v_2} = \left[\begin{array}{ccc}0\\1\\0\end{array}\right],\bold{v_3} = \left[\begin{array}{ccc}0\\0\\1\end{array}\right]}}[/tex]
[tex]\text{solusi lebih umumnya :}\\\\\Huge{\boxed{\boxed{\bold{v_2} = \left[\begin{array}{ccc}0\\y_2\\0\end{array}\right],\bold{v_3} = \left[\begin{array}{ccc}0\\0\\z_3\end{array}\right]}}[/tex]
[answer.2.content]
