Jika \( \begin{pmatrix} 1 & 1 \\ 2 & 1 \end{pmatrix} P \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \) dan \( \begin{pmatrix} 1 & 1 \\ 2 & 1 \end{pmatrix} P \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix} \), maka \( \det P = \cdots \)
- -3
- -2
- 1
- 2
- 3
(SBMPTN 2016 MATDAS)
Pembahasan:
Ingat bahwa jika \( AX = B \) maka \( X = A^{-1} B \) sehingga kita peroleh:
\begin{aligned} \begin{pmatrix} 1 & 1 \\ 2 & 1 \end{pmatrix} P \begin{pmatrix} 0 \\ 1 \end{pmatrix} &= \begin{pmatrix} 1 \\ 2 \end{pmatrix} \\[8pt] P \begin{pmatrix} 0 \\ 1 \end{pmatrix} &= \begin{pmatrix} 1 & 1 \\ 2 & 1 \end{pmatrix}^{-1} \begin{pmatrix} 1 \\ 2 \end{pmatrix} \\[8pt] P \begin{pmatrix} 0 \\ 1 \end{pmatrix} &= \frac{1}{1-2} \begin{pmatrix} 1 & -1 \\ -2 & 1 \end{pmatrix}\begin{pmatrix} 1 \\ 2 \end{pmatrix} \\[8pt] P \begin{pmatrix} 0 \\ 1 \end{pmatrix} &= \frac{1}{-1} \cdot \begin{pmatrix} -1 \\ 0 \end{pmatrix} \\[8pt] P \begin{pmatrix} 0 \\ 1 \end{pmatrix} &= \begin{pmatrix} 1 \\ 0 \end{pmatrix} \end{aligned}
Misalkan \( P = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), maka
\begin{aligned} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} &= \begin{pmatrix} 1 \\ 0 \end{pmatrix} \\[8pt] \begin{pmatrix} b \\ d \end{pmatrix} &= \begin{pmatrix} 1 \\ 0 \end{pmatrix} \\[8pt] b = 1 \ &\text{dan} \ d = 0 \end{aligned}
\begin{aligned} \begin{pmatrix} 1 & 1 \\ 2 & 1 \end{pmatrix} P \begin{pmatrix} 1 \\ 1 \end{pmatrix} &= \begin{pmatrix} 2 \\ 1 \end{pmatrix} \\[8pt] P \begin{pmatrix} 1 \\ 1 \end{pmatrix} &= \begin{pmatrix} 1 & 1 \\ 2 & 1 \end{pmatrix}^{-1} \begin{pmatrix} 2 \\ 1 \end{pmatrix} \\[8pt] P \begin{pmatrix} 1 \\ 1 \end{pmatrix} &= \frac{1}{1-2} \begin{pmatrix} 1 & -1 \\ -2 & 1 \end{pmatrix} \begin{pmatrix} 2 \\ 1 \end{pmatrix} \\[8pt] P \begin{pmatrix} 1 \\ 1 \end{pmatrix} &= \frac{1}{-1} \cdot \begin{pmatrix} 1 \\ -3 \end{pmatrix} \\[8pt] P \begin{pmatrix} 1 \\ 1 \end{pmatrix} &= \begin{pmatrix} -1 \\ 3 \end{pmatrix} \end{aligned}
\begin{aligned} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} &= \begin{pmatrix} -1 \\ 3 \end{pmatrix} \\[8pt] \begin{pmatrix} a+b \\ c+d \end{pmatrix} &= \begin{pmatrix} -1 \\ 3 \end{pmatrix} \\[8pt] a + b = -1 \Leftrightarrow a + 1 &= -1 \\[8pt] a &= -2 \\[8pt] c+d = 3 \Leftrightarrow c+0 &= 3 \\[8pt] c &= 3 \end{aligned}
Dengan demikian, determinan matriks P, yaitu:
\begin{aligned} P &= \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} -2 & 1 \\ 3 & 0 \end{pmatrix} \\[8pt] |P| &= \begin{vmatrix} -2 & 1 \\ 3 & 0 \end{vmatrix} = -2 \cdot 0 - 1 \cdot 3 \\[8pt] &= -3 \end{aligned}
Jawaban A.