Tentukan himpunan penyelesaian dari sistem persamaan 2x - Y - 9 = 0 dan x + 3Y - 1 = 0 menggunakan invers matriks.
Terimakasih^^
[tex] \: [/tex]
Terimakasih^^
[tex] \: [/tex]
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Jawaban:
![\displaystyle\left[\begin{array}{ccc}2&-1\\1&3\end{array}\right] \left[\begin{array}{ccc}x\\y\end{array}\right] =\left[\begin{array}{ccc}9\\1\end{array}\right] \\\\\left[\begin{array}{ccc}x\\y\end{array}\right] =\left[\begin{array}{ccc}2&-1\\1&3\end{array}\right]^{-1}\left[\begin{array}{ccc}9\\1\end{array}\right] \\\\\left[\begin{array}{ccc}x\\y\end{array}\right] =\frac{1}{ad-bc}\left[\begin{array}{ccc}d&c\\b&a\end{array}\right]\left[\begin{array}{ccc}9\\1\end{array}\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%26-1%5C%5C1%263%5Cend%7Barray%7D%5Cright%5D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx%5C%5Cy%5Cend%7Barray%7D%5Cright%5D%20%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D9%5C%5C1%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%5C%5C%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx%5C%5Cy%5Cend%7Barray%7D%5Cright%5D%20%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%26-1%5C%5C1%263%5Cend%7Barray%7D%5Cright%5D%5E%7B-1%7D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D9%5C%5C1%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%5C%5C%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx%5C%5Cy%5Cend%7Barray%7D%5Cright%5D%20%3D%5Cfrac%7B1%7D%7Bad-bc%7D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dd%26c%5C%5Cb%26a%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D9%5C%5C1%5Cend%7Barray%7D%5Cright%5D "\displaystyle\left[\begin{array}{ccc}2&-1\\1&3\end{array}\right] \left[\begin{array}{ccc}x\\y\end{array}\right] =\left[\begin{array}{ccc}9\\1\end{array}\right] \\\\\left[\begin{array}{ccc}x\\y\end{array}\right] =\left[\begin{array}{ccc}2&-1\\1&3\end{array}\right]^{-1}\left[\begin{array}{ccc}9\\1\end{array}\right] \\\\\left[\begin{array}{ccc}x\\y\end{array}\right] =\frac{1}{ad-bc}\left[\begin{array}{ccc}d&c\\b&a\end{array}\right]\left[\begin{array}{ccc}9\\1\end{array}\right]")
![\displaystyle\left[\begin{array}{ccc}x\\y\end{array}\right] =\frac{1}{2(3)-(-1)(1)}\left[\begin{array}{ccc}3&1\\-1&2\end{array}\right]\left[\begin{array}{ccc}9\\1\end{array}\right] \\\\\left[\begin{array}{ccc}x\\y\end{array}\right] =\frac{1}{6+1}\left[\begin{array}{ccc}3&1\\-1&2\end{array}\right]\left[\begin{array}{ccc}9\\1\end{array}\right]\\\\\left[\begin{array}{ccc}x\\y\end{array}\right] =\frac{1}{7}\left[\begin{array}{ccc}3(9)+1(1)\\-1(9)+2(1)\end{array}\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx%5C%5Cy%5Cend%7Barray%7D%5Cright%5D%20%3D%5Cfrac%7B1%7D%7B2%283%29-%28-1%29%281%29%7D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D3%261%5C%5C-1%262%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D9%5C%5C1%5Cend%7Barray%7D%5Cright%5D%20%5C%5C%5C%5C%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx%5C%5Cy%5Cend%7Barray%7D%5Cright%5D%20%3D%5Cfrac%7B1%7D%7B6%2B1%7D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D3%261%5C%5C-1%262%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D9%5C%5C1%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx%5C%5Cy%5Cend%7Barray%7D%5Cright%5D%20%3D%5Cfrac%7B1%7D%7B7%7D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D3%289%29%2B1%281%29%5C%5C-1%289%29%2B2%281%29%5Cend%7Barray%7D%5Cright%5D "\displaystyle\left[\begin{array}{ccc}x\\y\end{array}\right] =\frac{1}{2(3)-(-1)(1)}\left[\begin{array}{ccc}3&1\\-1&2\end{array}\right]\left[\begin{array}{ccc}9\\1\end{array}\right] \\\\\left[\begin{array}{ccc}x\\y\end{array}\right] =\frac{1}{6+1}\left[\begin{array}{ccc}3&1\\-1&2\end{array}\right]\left[\begin{array}{ccc}9\\1\end{array}\right]\\\\\left[\begin{array}{ccc}x\\y\end{array}\right] =\frac{1}{7}\left[\begin{array}{ccc}3(9)+1(1)\\-1(9)+2(1)\end{array}\right]")
![\displaystyle\left[\begin{array}{ccc}x\\y\end{array}\right] =\frac{1}{7}\left[\begin{array}{ccc}27+1\\-9+2\end{array}\right]\\\\\left[\begin{array}{ccc}x\\y\end{array}\right] =\frac{1}{7}\left[\begin{array}{ccc}28\\-7\end{array}\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx%5C%5Cy%5Cend%7Barray%7D%5Cright%5D%20%3D%5Cfrac%7B1%7D%7B7%7D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D27%2B1%5C%5C-9%2B2%5Cend%7Barray%7D%5Cright%5D%5C%5C%5C%5C%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx%5C%5Cy%5Cend%7Barray%7D%5Cright%5D%20%3D%5Cfrac%7B1%7D%7B7%7D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D28%5C%5C-7%5Cend%7Barray%7D%5Cright%5D "\displaystyle\left[\begin{array}{ccc}x\\y\end{array}\right] =\frac{1}{7}\left[\begin{array}{ccc}27+1\\-9+2\end{array}\right]\\\\\left[\begin{array}{ccc}x\\y\end{array}\right] =\frac{1}{7}\left[\begin{array}{ccc}28\\-7\end{array}\right]")
![\displaystyle\sf Hp=\left[\frac{28}{7}, -\frac{7}{7}\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Csf%20Hp%3D%5Cleft%5B%5Cfrac%7B28%7D%7B7%7D%2C%20-%5Cfrac%7B7%7D%7B7%7D%5Cright%5D "\displaystyle\sf Hp=\left[\frac{28}{7}, -\frac{7}{7}\right]")
Hp = {4, -1}
Penjelasan:
Tentukan Hp
2x-y-9 = 0
x+3y-1 = 0
Diubah menjadi
2x-y = 9
x+3y = 1
Dengan matriks
Hp = {4, -1} ✔️
xcvi