Invers matriks C ditulis C⁻¹, apabila dikalikan dengan matriks C akan menghasilkan matriks Identitas. Invers dari matriks C adalah C⁻¹ [tex]=\begin{pmatrix}-\frac{1}{2}&2&-\frac{5}{2}\\ 1&-1&1\\ -\frac{1}{2}&0&\frac{1}{2}\end{pmatrix}[/tex].
Penjelasan dengan langkah-langkah
Invers dari matriks A yang mempunyai ordo 3 x 3
A⁻¹ = [tex]\frac{1}{det A} Adj A[/tex]
Penjelasan Soal:
Diketahui:
[tex]C = \begin{pmatrix}1&2&1\\ \:2&3&4\\ \:1&2&3\end{pmatrix}[/tex]
Ditanya:
invers matriks C
Jawab:
Determinan dengan metode saurus
[tex]|C| = \begin{vmatrix}1&2&1\\ \:\:2&3&4\\ \:\:1&2&3\end{vmatrix}\begin{matrix} \\\end{matrix}\left.\begin{matrix}1&2\\ 2&3\\ 1&2 \\\end{matrix}\right|[/tex]
= ((1)(3)(3) + (2)(4)(1) + (1)(2)(2)) - ((1)(3)(1) + (1)(4)(2) + (2)(2)(3))
= (9 + 8 +4) - (3 + 8 + 12)
= 21 - 23
= -2
Minor matriks
M₁₁ = determinan matriks ordo 2x2 yang diperoleh dari menghilangkan elemen-elemen pada baris ke 1 kolom ke 1 pada matriks ordo 3x3
M₁₁ = [tex]\begin{vmatrix}3&4\\ 2&3\end{vmatrix}[/tex] = 3.3 - 4.2 = 9 - 8 = 1
M₁₂ = [tex]\begin{vmatrix}2&4\\ 1&3\end{vmatrix}[/tex] = 6 - 4 = 2
M₁₃ = [tex]\begin{vmatrix}2&3\\ 1&2\end{vmatrix}[/tex] = 4 - 3 = 1
M₂₁ = [tex]\begin{vmatrix}2&1\\ 2&3\end{vmatrix}[/tex] = 6 - 2 = 4
M₂₂ = [tex]\begin{vmatrix}1&1\\ 1&3\end{vmatrix}[/tex] = 3 - 1 = 2
M₂₃ = [tex]\begin{vmatrix}1&2\\ 1&2\end{vmatrix}[/tex] = 2 - 2 = 0
M₃₁ = [tex]\begin{vmatrix}2&1\\ 3&4\end{vmatrix}[/tex] = 8 - 3 = 5
M₃₂ = [tex]\begin{vmatrix}1&1\\ 2&4\end{vmatrix}[/tex] = 4 -2 = 2
M₃₃ =[tex]\begin{vmatrix}1&2\\ 2&3\end{vmatrix}[/tex] = 3 - 4 = -1
Minor matriks C = [tex]\begin{pmatrix}1&2&1\\ 4&2&0\\ 5&2&-1\end{pmatrix}[/tex]
Kofaktor matriks C
Kofaktor matriks C baris pertama kolom pertama, berarti i = 1 dan j = 1
Kᵢⱼ = (-1) Mⁱ⁺ʲ · Mᵢⱼ
K₁₁ = (-1)²· 1 = 1
K₁₂ = (-1)³· 2 = -2
K₁₃ = (-1)⁴· 1 = 1
K₂₁ = (-1)³· 4 = -4
K₂₂ = (-1)⁴· 2 = 2
K₂₃ = (-1)⁵· 0 = 0
K₃₁ = (-1)⁴· 5 = 5
K₃₂ = (-1)⁵· 2 = -2
K₃₃ = (-1)⁶· -1 = -1
Kofaktor matriks C = [tex]\begin{pmatrix}1&-2&1\\ -4&2&0\\ 5&-2&-1\end{pmatrix}[/tex]
Adjoin matriks C
Adjoin matriks C = transpose kofaktor matriks C
Adj C = [tex]\begin{pmatrix}1&-4&5\\ -2&2&-2\\ 1&0&-1\end{pmatrix}[/tex]
Invers matriks C
C⁻¹ = [tex]\frac{1}{det C} Adj C[/tex]
[tex]= \frac{1}{-2}\begin{pmatrix}1&-4&5\\ -2&2&-2\\ 1&0&-1\end{pmatrix}[/tex]
[tex]=\begin{pmatrix}-\frac{1}{2}&2&-\frac{5}{2}\\ 1&-1&1\\ -\frac{1}{2}&0&\frac{1}{2}\end{pmatrix}[/tex]
Pelajari lebih lanjut
Materi invers matriks ordo 3x3 https://brainly.co.id/tugas/5318631
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