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SMK Teknologi dan Rekayasa

Soal 44 Operasi Matriks


Jika\ A=\begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}\ dan\ B=\begin{pmatrix} 5 & 4 \\ 2 & 1 \end{pmatrix},\ maka\ determinan\ (A.B)^{-1}\ adalah\ ....
a.\ -\frac{1}{4}\ \ \ \ \ \ \ \ \ \ b.\ \ \frac{1}{4}\ \ \ \ \ \ \ \ \ \ c.\ -\frac{1}{3}\ \ \ \ \ \ \ \ \ \ d.\ \ \frac{1}{3}\ \ \ \ \ \ \ \ \ \ e.\ -\frac{1}{2}
Jawaban: c

\begin{array}{rcl}A.B&=&\begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}\begin{pmatrix} 5 & 4 \\ 2 & 1 \end{pmatrix}\\\\&=& \begin{pmatrix} 2\times 5\ +\ 5\times 2 & & 2\times 4\ +\ 5\times 1 \\ 1\times 5\ +\ 3\times 2 & & 1\times 4\ +\ 3\times 1 \end{pmatrix}\\\\&=&\begin{pmatrix} 10+10 & 8+5 \\ 5+6 & 4+3 \end{pmatrix}\\\\&=&\begin{pmatrix} 20 & 13 \\ 11 & 7 \end{pmatrix}\end{array}
\begin{array}{rcl}Determinan\ A.B\ adalah\ \begin{vmatrix} A.B \end{vmatrix}&=&20.7-13.11\\&=&140-143\\&=&-3\end{array}
\begin{array}{rcl}Determinan\ (A.B)^{-1}\ adalah\ \begin{vmatrix} (A.B)^{-1} \end{vmatrix}&=&\frac{1}{\begin{vmatrix} A.B \end{vmatrix}}\\\\&=&\frac{1}{-3}\\\\&=&-\frac{1}{3}\end{array}

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