2010 Pembahasan Persamaan Matriks UTUL UGM Matematika Ipa

Soal yang Akan Dibahas
Diketahui matriks $ X = \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] $
dan $ P = \left[ \begin{matrix} 1 & 4 \\ 2 & 6 \end{matrix} \right] $ , serta $ PX = P^{-1} $. Nilai $ a + b + c + d = .... $
A). $\frac{11}{4} \, $ B). $ 95 \, $ C). $\frac{95}{4} \, $ D). $-\frac{95}{4} \, $ E). $-\frac{11}{4} \, $

$\spadesuit $ Konsep Dasar Matriks
*). Invers matriks
$ A = \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \rightarrow A^{-1} = \frac{1}{ad-bc}\left[ \begin{matrix} d & -b \\ -c & a \end{matrix} \right] $
*). Persamaan matriks : $ AX=B \rightarrow X = B.A^{-1} $

$\clubsuit $ Pembahasan
*). Menentukan invers matriks P :
$\begin{align} P & = \left[ \begin{matrix} 1 & 4 \\ 2 & 6 \end{matrix} \right] \\ P^{-1} & = \frac{1}{1.6 - 4.2} \left[ \begin{matrix} 6 & -4 \\ -2 & 1 \end{matrix} \right] \\ & = \frac{1}{-2} \left[ \begin{matrix} 6 & -4 \\ -2 & 1 \end{matrix} \right] \end{align} $
*). Menentukan matriks $ X $ :
$\begin{align} PX & = P^{-1} \\ X & = P^{-1} . P^{-1} \\ & = \frac{1}{-2} \left[ \begin{matrix} 6 & -4 \\ -2 & 1 \end{matrix} \right] . \frac{1}{-2} \left[ \begin{matrix} 6 & -4 \\ -2 & 1 \end{matrix} \right] \\ & = \frac{1}{-2} .\frac{1}{-2}\left[ \begin{matrix} 6 & -4 \\ -2 & 1 \end{matrix} \right] . \left[ \begin{matrix} 6 & -4 \\ -2 & 1 \end{matrix} \right] \\ & = \frac{1}{4}\left[ \begin{matrix} 44 & -28 \\ -14 & 9 \end{matrix} \right] \\ \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] & = \left[ \begin{matrix} 11 & -7 \\ -\frac{7}{2} & \frac{9}{4} \end{matrix} \right] \end{align} $
*). Menentukan nilai $ a + b + c + d $ :
$ \begin{align} a + b + c + d & = 11 + (-7) + (-\frac{7}{2}) + \frac{9}{4} \\ & = \frac{44 - 28 - 14 + 9}{4} \\ & = \frac{11}{4} \end{align} $
Jadi, nilai $ a + b + c + d = \frac{11}{4} . \, \heartsuit $



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