Tentukan Nilai Eigen dan Vektor eigen dari matriks berikut. sertakan cara trima kasih
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Nilai eigennya adalah 4 atau -2 dan vektor eigen untuk nilai eigen 4 adalah![\left[\begin{array}{cc}6\\9\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D6%5C%5C9%5Cend%7Barray%7D%5Cright%5D "\left[\begin{array}{cc}6\\9\end{array}\right]")
Diberikan matriks
Maka, untuk mencari nilai eigennya ingat kita dapat menggunakan persamaan karakteristik :
det {A - λI) = 0
• cari nilai matriks A - λI
A - λI =![\left[\begin{array}{cc}4&0\\9&-2\end{array}\right] - \lambda \left[\begin{array}{cc}1&0\\0&1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D4%260%5C%5C9%26-2%5Cend%7Barray%7D%5Cright%5D%20-%20%5Clambda%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%260%5C%5C0%261%5Cend%7Barray%7D%5Cright%5D "\left[\begin{array}{cc}4&0\\9&-2\end{array}\right] - \lambda \left[\begin{array}{cc}1&0\\0&1\end{array}\right]")
A - λI =![\left[\begin{array}{cc}4&0\\9&-2\end{array}\right] - \left[\begin{array}{cc}\lambda&0\\0&\lambda\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D4%260%5C%5C9%26-2%5Cend%7Barray%7D%5Cright%5D%20-%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%5Clambda%260%5C%5C0%26%5Clambda%5Cend%7Barray%7D%5Cright%5D "\left[\begin{array}{cc}4&0\\9&-2\end{array}\right] - \left[\begin{array}{cc}\lambda&0\\0&\lambda\end{array}\right]")
A - λI =![\left[\begin{array}{cc}4 - \lambda&0\\9&-2 - \lambda\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D4%20-%20%5Clambda%260%5C%5C9%26-2%20-%20%5Clambda%5Cend%7Barray%7D%5Cright%5D "\left[\begin{array}{cc}4 - \lambda&0\\9&-2 - \lambda\end{array}\right]")
• cari determinan matriks A - λI
det (A - λI) = ad - bc
det (A - λI) = (4 - λ)(-2 - λ) - (0)(9)
det (A - λI) = -8 - 4λ + 2λ + λ² - 0
det (A - λI) = λ² - 2λ - 8
ingat persamaan karakteristik, det (A - λI) = 0
det (A - λI) = λ² - 2λ - 8
0 = λ² - 2λ - 8
(λ - 4)(λ + 2) = 0
λ = 4 V λ = -2
Maka, nilai eigennya adalah 4 atau -2
Untuk mencari vektor eigen, gunakan persamaan (A - Iλ) v = 0
(A - Iλ) v = 0
dimana v adalah vektor eigen![\left[\begin{array}{cc}\text{v}_1\\\text{v}_2\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%5Ctext%7Bv%7D_1%5C%5C%5Ctext%7Bv%7D_2%5Cend%7Barray%7D%5Cright%5D "\left[\begin{array}{cc}\text{v}_1\\\text{v}_2\end{array}\right]")
• untuk nilai eigen 4
Didapatkan persamaan linier
9v₁ - 6v₂ = 0
9v₁ = 6v₂
Maka, vektor eigen v =![\left[\begin{array}{cc}\text{v}_1\\\text{v}_2\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D%5Ctext%7Bv%7D_1%5C%5C%5Ctext%7Bv%7D_2%5Cend%7Barray%7D%5Cright%5D "\left[\begin{array}{cc}\text{v}_1\\\text{v}_2\end{array}\right]")
= ![\left[\begin{array}{cc}6\\9\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D6%5C%5C9%5Cend%7Barray%7D%5Cright%5D "\left[\begin{array}{cc}6\\9\end{array}\right]")
PEMBUKTIAN :
Av = λv
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Pelajari lebih lanjut
Materi tentang Nilai dan Vektor Eigen
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DETIL JAWABAN
Kelas :
Mapel : Aljabar Linier
Bab : Nilai Eigen dan Vektor Eigen
Kode : 12.2.3
Kata Kunci : Nilai Eigen, Vektor Eigen, Matriks, Matriks Identitas, Matriks Augmentasi
Mapel : Math
Topik : Vektor
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