२ अज्ञात राशि वाले २ रैखिक समीकरणों का निकाय संपादित करें माना दो अज्ञात राशि से युक्त दो रैखिक समीकरण ये हैं:
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{\displaystyle {\begin{cases}a{\color {blue}x}+b{\color {blue}y}={\color {red}e}\\c{\color {blue}x}+d{\color {blue}y}={\color {red}f}\end{cases}}}
इनका मैट्रिक्स निरूपण यह है:
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{\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}{\begin{bmatrix}{\color {blue}x}\\{\color {blue}y}\end{bmatrix}}={\begin{bmatrix}{\color {red}e}\\{\color {red}f}\end{bmatrix}}}
क्रैमर का नियम लगाकर
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{\displaystyle x}
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{\displaystyle y}
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{\displaystyle x={\frac {\begin{vmatrix}\color {red}{e}&b\\\color {red}{f}&d\end{vmatrix}}{\begin{vmatrix}a&b\\c&d\end{vmatrix}}}={\frac {{\color {red}e}d-b{\color {red}f}}{ad-bc}};\quad y={\frac {\begin{vmatrix}a&\color {red}{e}\\c&\color {red}{f}\end{vmatrix}}{\begin{vmatrix}a&b\\c&d\end{vmatrix}}}={\frac {a{\color {red}f}-{\color {red}e}c}{ad-bc}}}
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{\displaystyle 3x+1y=9\,}
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{\displaystyle 2x+3y=13\,}
इनको मैट्रिक्स रूप में लिखने पर:
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{\displaystyle {\begin{bmatrix}3&1\\2&3\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}={\begin{bmatrix}9\\13\end{bmatrix}}}
क्रैमर नियम से x और y का मान यह है:
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{\displaystyle x={\frac {\begin{vmatrix}9&1\\13&3\end{vmatrix}}{\begin{vmatrix}3&1\\2&3\end{vmatrix}}}={9*3-1*13 \over 3*3-1*2}=2}
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{\displaystyle y={\frac {\begin{vmatrix}3&9\\2&13\end{vmatrix}}{\begin{vmatrix}3&1\\2&3\end{vmatrix}}}={3*13-9*2 \over 3*3-1*2}=3}
माना मैट्रिक्स रूप में निरूपित 3x3 रैखिक समीकरण निकाय यह है:
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{\displaystyle {\begin{cases}a{\color {blue}x}+b{\color {blue}y}+c{\color {blue}z}={\color {black}j}\\d{\color {blue}x}+e{\color {blue}y}+f{\color {blue}z}={\color {black}k}\\g{\color {blue}x}+h{\color {blue}y}+i{\color {blue}z}={\color {black}l}\end{cases}}}
इसका हल यह है:
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{\displaystyle {\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}}{\begin{bmatrix}{\color {blue}x}\\{\color {blue}y}\\{\color {blue}z}\end{bmatrix}}={\begin{bmatrix}{\color {red}j}\\{\color {red}k}\\{\color {red}l}\end{bmatrix}}}
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{\displaystyle x}
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{\displaystyle x={\frac {\begin{vmatrix}{\color {red}j}&b&c\\{\color {red}k}&e&f\\{\color {red}l}&h&i\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}};\quad y={\frac {\begin{vmatrix}a&{\color {red}j}&c\\d&{\color {red}k}&f\\g&{\color {red}l}&i\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}},\quad z={\frac {\begin{vmatrix}a&b&{\color {red}j}\\d&e&{\color {red}k}\\g&h&{\color {red}l}\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}}}
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{\displaystyle 3x+2y+1z=1\,}
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{\displaystyle 2x+0y+1z=2\,}
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{\displaystyle -1x+1y+2z=4\,}
मैट्रिक्स रूप में लिखने पर:
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{\displaystyle {\begin{bmatrix}3&2&1\\2&0&1\\-1&1&2\end{bmatrix}}{\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}1\\2\\4\end{bmatrix}}}
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{\displaystyle x,y{\text{ y }}z}
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{\displaystyle x={\frac {\begin{vmatrix}1&2&1\\2&0&1\\4&1&2\end{vmatrix}}{\begin{vmatrix}3&2&1\\2&0&1\\-1&1&2\end{vmatrix}}};\quad y={\frac {\begin{vmatrix}3&1&1\\2&2&1\\-1&4&2\end{vmatrix}}{\begin{vmatrix}3&2&1\\2&0&1\\-1&1&2\end{vmatrix}}};\quad z={\frac {\begin{vmatrix}3&2&1\\2&0&2\\-1&1&4\end{vmatrix}}{\begin{vmatrix}3&2&1\\2&0&1\\-1&1&2\end{vmatrix}}}}
![{\displaystyle \mathbf {A} _{j}=\left[{\begin{array}{llllllll}a_{1,1}&\cdots &a_{1,j-1}&b_{1}&a_{1,j+1}&\cdots &a_{1,n}\\a_{2,1}&\cdots &a_{2,j-1}&b_{2}&a_{2,j+1}&\cdots &a_{2,n}\\\\\vdots &&&\ddots &&&\vdots \\\\a_{n-1,1}&\cdots &a_{n-1,j-1}&b_{n-1}&a_{n-1,j+1}&\cdots &a_{n-1,n}\\a_{n,1}&\cdots &a_{n,j-1}&b_{n}&a_{n,j+1}&\cdots &a_{n,n}\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a5805b695ed51423b97f058e8298ea58f3ee624)
