TUGAS 3 CARA MENYELSAIKAN MATRIKS ORDO 3X3
Carilah invers matriks 3×3 yaitu A =
![\left [\begin{array}{rrr} 1& 2& 3\\ 2& 5& 3\\ 1& 0& 8 \end{array} \right ]](https://s0.wp.com/latex.php?latex=%5Cleft+%5B%5Cbegin%7Barray%7D%7Brrr%7D+1%26+2%26+3%5C%5C+2%26+5%26+3%5C%5C+1%26+0%26+8+%5Cend%7Barray%7D+%5Cright+%5D&bg=ffffff&fg=666666&s=0 "\left [\begin{array}{rrr} 1& 2& 3\\ 2& 5& 3\\ 1& 0& 8 \end{array} \right ]")
Penyelesaian :
Susun matriks sedemikian sehingga seperti dibawah ini.
![\left[ \left.\begin{matrix} 1& 2& 3\\ 2& 5& 3\\ 1& 0& 8 \end{matrix}\right| \begin {array}{rrr} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end {array}\right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%5Cleft.%5Cbegin%7Bmatrix%7D+1%26+2%26+3%5C%5C+2%26+5%26+3%5C%5C+1%26+0%26+8+%5Cend%7Bmatrix%7D%5Cright%7C+%5Cbegin+%7Barray%7D%7Brrr%7D+1%26+0%26+0%5C%5C+0%26+1%26+0%5C%5C+0%26+0%26+1%5Cend+%7Barray%7D%5Cright%5D&bg=ffffff&fg=666666&s=0 "\left[ \left.\begin{matrix} 1& 2& 3\\ 2& 5& 3\\ 1& 0& 8 \end{matrix}\right| \begin {array}{rrr} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end {array}\right]")
Matriks disebelah kiri adalah matriks A dan sebelah kanan adalah matriks identitas. Kemudian lakukan Operasi Baris Elementer sedemikan sehingga matriks sebelah kiri menjadi matriks identitas dan matriks identitas (pada sebelah kanan) yang akan menjadi invers matriks tersebut.baris kedua : B2 + (-2B1) [artinya baris kedua dijumlahkan dengan -2 kali baris pertama]
- baris ketiga : B3 + (-B1) [artinya baris kedua dijumlahkan dengan -1 kali baris pertama]
- baris ketiga : B3 + 2B2 [artinya baris ketiga dijumlahkan dengan 2 kali baris kedua]
- baris ketiga : B3 x (-1) [artinya baris ketiga dikali dengan -1]
- baris kedua : B2 + 3B3 [artinya baris kedua dijumlahkan dengan 3 kali baris ketiga]baris pertama : B1 + (-3B3) [artinya baris pertama dijumlahkan dengan -3 kali baris ketiga]
- baris pertama : B1 + (-2B2) [artinya baris pertama dijumlahkan dengan -2 kali baris kedua]
![\left[ \left.\begin{matrix} 1& 2& 3\\ 0& 1& -3\\ 0& -2& 5 \end{matrix}\right| \begin {array}{rrr} 1& 0& 0\\ -2& 1& 0\\ -1& 0& 1\end {array}\right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%5Cleft.%5Cbegin%7Bmatrix%7D+1%26+2%26+3%5C%5C+0%26+1%26+-3%5C%5C+0%26+-2%26+5+%5Cend%7Bmatrix%7D%5Cright%7C+%5Cbegin+%7Barray%7D%7Brrr%7D+1%26+0%26+0%5C%5C+-2%26+1%26+0%5C%5C+-1%26+0%26+1%5Cend+%7Barray%7D%5Cright%5D&bg=ffffff&fg=666666&s=0 "\left[ \left.\begin{matrix} 1& 2& 3\\ 0& 1& -3\\ 0& -2& 5 \end{matrix}\right| \begin {array}{rrr} 1& 0& 0\\ -2& 1& 0\\ -1& 0& 1\end {array}\right]")
![\left[ \left.\begin{matrix} 1& 2& 3\\ 0& 1& -3\\ 0& 0& -1 \end{matrix}\right| \begin {array}{rrr} 1& 0& 0\\ -2& 1& 0\\ -5& 2& 1\end {array}\right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%5Cleft.%5Cbegin%7Bmatrix%7D+1%26+2%26+3%5C%5C+0%26+1%26+-3%5C%5C+0%26+0%26+-1+%5Cend%7Bmatrix%7D%5Cright%7C+%5Cbegin+%7Barray%7D%7Brrr%7D+1%26+0%26+0%5C%5C+-2%26+1%26+0%5C%5C+-5%26+2%26+1%5Cend+%7Barray%7D%5Cright%5D&bg=ffffff&fg=666666&s=0 "\left[ \left.\begin{matrix} 1& 2& 3\\ 0& 1& -3\\ 0& 0& -1 \end{matrix}\right| \begin {array}{rrr} 1& 0& 0\\ -2& 1& 0\\ -5& 2& 1\end {array}\right]")
![\left[ \left.\begin{matrix} 1& 2& 3\\ 0& 1& -3\\ 0& 0& 1 \end{matrix}\right| \begin {array}{rrr} 1& 0& 0\\ -2& 1& 0\\ 5& -2& -1\end {array}\right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%5Cleft.%5Cbegin%7Bmatrix%7D+1%26+2%26+3%5C%5C+0%26+1%26+-3%5C%5C+0%26+0%26+1+%5Cend%7Bmatrix%7D%5Cright%7C+%5Cbegin+%7Barray%7D%7Brrr%7D+1%26+0%26+0%5C%5C+-2%26+1%26+0%5C%5C+5%26+-2%26+-1%5Cend+%7Barray%7D%5Cright%5D&bg=ffffff&fg=666666&s=0 "\left[ \left.\begin{matrix} 1& 2& 3\\ 0& 1& -3\\ 0& 0& 1 \end{matrix}\right| \begin {array}{rrr} 1& 0& 0\\ -2& 1& 0\\ 5& -2& -1\end {array}\right]")
![\left[ \left.\begin{matrix} 1& 2& 0\\ 0& 1& 0\\ 0& 0& 1 \end{matrix}\right| \begin {array}{rrr} -14& 6& 3\\ 13& -5& -3\\ 5& -2& -1\end {array}\right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%5Cleft.%5Cbegin%7Bmatrix%7D+1%26+2%26+0%5C%5C+0%26+1%26+0%5C%5C+0%26+0%26+1+%5Cend%7Bmatrix%7D%5Cright%7C+%5Cbegin+%7Barray%7D%7Brrr%7D+-14%26+6%26+3%5C%5C+13%26+-5%26+-3%5C%5C+5%26+-2%26+-1%5Cend+%7Barray%7D%5Cright%5D&bg=ffffff&fg=666666&s=0 "\left[ \left.\begin{matrix} 1& 2& 0\\ 0& 1& 0\\ 0& 0& 1 \end{matrix}\right| \begin {array}{rrr} -14& 6& 3\\ 13& -5& -3\\ 5& -2& -1\end {array}\right]")
![\left [ \left.\begin{matrix} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{matrix}\right|\begin{matrix} -40& 16& 9\\ 13& -5& -3\\ 5& -2& -1 \end{matrix} \right ]](https://s0.wp.com/latex.php?latex=%5Cleft+%5B+%5Cleft.%5Cbegin%7Bmatrix%7D+1%26+0%26+0%5C%5C+0%26+1%26+0%5C%5C+0%26+0%26+1%5Cend%7Bmatrix%7D%5Cright%7C%5Cbegin%7Bmatrix%7D+-40%26+16%26+9%5C%5C+13%26+-5%26+-3%5C%5C+5%26+-2%26+-1+%5Cend%7Bmatrix%7D+%5Cright+%5D&bg=ffffff&fg=666666&s=0 "\left [ \left.\begin{matrix} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{matrix}\right|\begin{matrix} -40& 16& 9\\ 13& -5& -3\\ 5& -2& -1 \end{matrix} \right ]")
![\left[ \left.\begin{matrix} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1 \end{matrix}\right| \begin {array}{rrr} -40& 16& 9\\ 13& -5& -3\\ 5& -2& -1\end {array}\right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B+%5Cleft.%5Cbegin%7Bmatrix%7D+1%26+0%26+0%5C%5C+0%26+1%26+0%5C%5C+0%26+0%26+1+%5Cend%7Bmatrix%7D%5Cright%7C+%5Cbegin+%7Barray%7D%7Brrr%7D+-40%26+16%26+9%5C%5C+13%26+-5%26+-3%5C%5C+5%26+-2%26+-1%5Cend+%7Barray%7D%5Cright%5D&bg=ffffff&fg=666666&s=0 "\left[ \left.\begin{matrix} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1 \end{matrix}\right| \begin {array}{rrr} -40& 16& 9\\ 13& -5& -3\\ 5& -2& -1\end {array}\right]")
Karena matriks kiri sudah terbentuk menjadi matriks identitas, maka invers dari matriks A adalah A-1 = ![\left [\begin{matrix} -40& 16& 9\\ 13& -5& -3\\ 5& -2& -1 \end{matrix} \right ]](https://s0.wp.com/latex.php?latex=%5Cleft+%5B%5Cbegin%7Bmatrix%7D+-40%26+16%26+9%5C%5C+13%26+-5%26+-3%5C%5C+5%26+-2%26+-1+%5Cend%7Bmatrix%7D+%5Cright+%5D&bg=ffffff&fg=666666&s=0 "\left [\begin{matrix} -40& 16& 9\\ 13& -5& -3\\ 5& -2& -1 \end{matrix} \right ]")
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