Invers dari matriks A=(-1 3 -2 2 -1 1 -2 3 1)

$\begin{pmatrix} \frac{1}{4} & \frac{9}{16} & -\frac{1}{16} \\ \frac{1}{4} & \frac{5}{16} & \frac{3}{16} \\ -\frac{1}{4} & \frac{3}{16} & \frac{5}{16} \end{pmatrix}$

Pembahasan

Untuk mencari invers dari matriks $A = \begin{pmatrix} -1 & 3 & -2 \\ 2 & -1 & 1 \\ -2 & 3 & 1 \end{pmatrix}$, kita perlu mengikuti langkah-langkah berikut:

1.  **Hitung Determinan (det(A))**: 
    det(A) = $-1((-1)(1) - (1)(3)) - 3((2)(1) - (1)(-2)) + (-2)((2)(3) - (-1)(-2))$
    det(A) = $-1(-1 - 3) - 3(2 + 2) - 2(6 - 2)$
    det(A) = $-1(-4) - 3(4) - 2(4)$
    det(A) = $4 - 12 - 8$
    det(A) = $-16$
    Karena determinan tidak nol, matriks A memiliki invers.

2.  **Cari Matriks Adjoin (adj(A))**: Matriks adjoin adalah transpose dari matriks kofaktor.
    *   **Kofaktor C11:** $(-1)^{1+1} egin{vmatrix} -1 & 1 \ 3 & 1 
vert = 1(-1 - 3) = -4$
    *   **Kofaktor C12:** $(-1)^{1+2} egin{vmatrix} 2 & 1 \ -2 & 1 
vert = -1(2 - (-2)) = -4$
    *   **Kofaktor C13:** $(-1)^{1+3} egin{vmatrix} 2 & -1 \ -2 & 3 
vert = 1(6 - 2) = 4$
    *   **Kofaktor C21:** $(-1)^{2+1} egin{vmatrix} 3 & -2 \ 3 & 1 
vert = -1(3 - (-6)) = -9$
    *   **Kofaktor C22:** $(-1)^{2+2} egin{vmatrix} -1 & -2 \ -2 & 1 
vert = 1(-1 - 4) = -5$
    *   **Kofaktor C23:** $(-1)^{2+3} egin{vmatrix} -1 & 3 \ -2 & 3 
vert = -1(-3 - (-6)) = -3$
    *   **Kofaktor C31:** $(-1)^{3+1} egin{vmatrix} 3 & -2 \ -1 & 1 
vert = 1(3 - 2) = 1$
    *   **Kofaktor C32:** $(-1)^{3+2} egin{vmatrix} -1 & -2 \ 2 & 1 
vert = -1(-1 - (-4)) = -3$
    *   **Kofaktor C33:** $(-1)^{3+3} egin{vmatrix} -1 & 3 \ 2 & -1 
vert = 1(1 - 6) = -5$

    Matriks Kofaktor = $\begin{pmatrix} -4 & -4 & 4 \\ -9 & -5 & -3 \\ 1 & -3 & -5 \end{pmatrix}$

    Matriks Adjoin (transpose dari matriks kofaktor) = $\begin{pmatrix} -4 & -9 & 1 \\ -4 & -5 & -3 \\ 4 & -3 & -5 \end{pmatrix}$

3.  **Hitung Invers (A⁻¹)**:
    $A^{-1} = \frac{1}{det(A)} adj(A)$
    $A^{-1} = \frac{1}{-16} \begin{pmatrix} -4 & -9 & 1 \\ -4 & -5 & -3 \\ 4 & -3 & -5 \end{pmatrix}$
    $A^{-1} = \begin{pmatrix} \frac{-4}{-16} & \frac{-9}{-16} & \frac{1}{-16} \\ \frac{-4}{-16} & \frac{-5}{-16} & \frac{-3}{-16} \\ \frac{4}{-16} & \frac{-3}{-16} & \frac{-5}{-16} \end{pmatrix}$
    $A^{-1} = \begin{pmatrix} \frac{1}{4} & \frac{9}{16} & -\frac{1}{16} \\ \frac{1}{4} & \frac{5}{16} & \frac{3}{16} \\ -\frac{1}{4} & \frac{3}{16} & \frac{5}{16} \end{pmatrix}$