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SMK Teknologi dan Rekayasa

Soal 43 Operasi Matriks


Diketahui\ matriks\ A=\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}\ dan\ B=\begin{pmatrix} 10 & 20 \\ 20 & 30 \end{pmatrix},\\maka\ matriks\ A^{-1}.B\ adalah\ ....\\\\a.\ \ \begin{pmatrix} 5 & 2 \\ 5 & 4 \end{pmatrix}\ \ \ \ \ \ \ \ \ \ b.\ \ \begin{pmatrix} 5 & 4 \\ 2 & 5 \end{pmatrix}\ \ \ \ \ \ \ \ \ \ c.\ \ \begin{pmatrix} 5 & 2 \\ 4 & 5 \end{pmatrix}\\\\\\d.\ \ \begin{pmatrix} 2 & 4 \\ 5 & 5 \end{pmatrix}\ \ \ \ \ \ \ \ \ \ e.\ \ \begin{pmatrix} 2 & 5 \\ 4 & 5 \end{pmatrix}
Jawaban: e

A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\ \ \to \ \ A^{-1}=\frac{1}{a.d-b.c}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\begin{array}{rcl}A=\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}\ \ \to \ \ A^{-1}&=&\frac{1}{3.4-1.2}\begin{pmatrix} 4 & -1 \\ -2 & 3 \end{pmatrix}\\\\&=&\frac{1}{12-2}\begin{pmatrix} 4 & -1 \\ -2 & 3 \end{pmatrix}\\\\&=&\frac{1}{10}\begin{pmatrix} 4 & -1 \\ -2 & 3 \end{pmatrix}\\\\&=&\begin{pmatrix} \frac{2}{5} & -\frac{1}{10} \\ -\frac{1}{5} & \frac{3}{10} \end{pmatrix}\end{array}

\begin{array}{rcl}A^{-1}.B&=&\begin{pmatrix} \frac{2}{5} & -\frac{1}{10} \\ -\frac{1}{5} & \frac{3}{10} \end{pmatrix}\begin{pmatrix} 10 & 20 \\ 20 & 30 \end{pmatrix}\\\\&=&\begin{pmatrix} \frac{2}{5}\ \times 10\ +\ \left (-\frac{1}{10}\ \times 20 \right ) & & \frac{2}{5}\ \times 20\ +\ \left (-\frac{1}{10}\ \times 30 \right ) \\ -\frac{1}{5}\ \times 10\ +\ \frac{3}{10}\ \times 20 & & -\frac{1}{5}\ \times 20\ +\ \frac{3}{10}\ \times 30 \end{pmatrix}\\\\&=&\begin{pmatrix} 4+(-2) & 8+(-3) \\ -2+6 & -4+9 \end{pmatrix}\\\\&=&\begin{pmatrix} 2 & 5 \\ 4 & 5 \end{pmatrix}\end{array}

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One thought on “Soal 43 Operasi Matriks

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