Diketahui matriks \( A = \begin{pmatrix} 1 & -2 \\ 2 & 5 \end{pmatrix} \). Nilai \( k \) yang memenuhi \( k \cdot \det(A^T) = \det(A^{-1}) \) adalah…
- 81
- 9
- 1
- 1/9
- 1/81
(SIMAK UI 2010 MATDAS)
Pembahasan:
\begin{aligned} A^T = \begin{pmatrix} 1 & -2 \\ 2 & 5 \end{pmatrix}^T &= \begin{pmatrix} 1 & 2 \\ -2 & 5 \end{pmatrix} \\[8pt] A^{-1} = \begin{pmatrix} 1 & -2 \\ 2 & 5 \end{pmatrix}^{-1} &= \frac{1}{5+4} \begin{pmatrix} 5 & -2 \\ 2 & 1 \end{pmatrix} \\[8pt] A^{-1} &= \begin{pmatrix} \frac{5}{9} & -\frac{2}{9} \\[8pt] \frac{2}{9} & \frac{1}{9} \end{pmatrix} \\[8pt] k \cdot \det(A^T) &= \det(A^{-1}) \\[8pt] k (5+4) &= \frac{5}{9} \cdot \frac{1}{9}-\left( -\frac{2}{9} \right) \cdot \frac{2}{9} \\[8pt] 9k &= \frac{5}{81}+\frac{4}{81} \Leftrightarrow 9k = \frac{1}{9} \\[8pt] k &= \frac{1}{81} \end{aligned}
Jawaban E.