Word Count : 1424

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दिए गए समीकरण-निकाय को कीलकन (Pivoting) के साथ गाउस विलोपन विधि (Gaussian Elimination Method) का उपयोग करके हल करेंगे। समीकरण-निकाय निम्नलिखित है: \[ \begin{aligned} 2x_1 + 3x_2 + 4x_3 + x_4 &= 3 \quad \text{(1)} \\ x_1 + 2x_2 + x_4 &= 2 \quad \text{(2)} \\ 2x_1 + 3x_2 + x_3 - x_4 &= 1 \quad \text{(3)} \\ x_1 - 2x_2 - x_3 + 4x_4 &= 5 \quad \text{(4)} \end{aligned} \] ### चरण 1: समीकरण-निकाय को आव्यूह रूप में लिखें समीकरण-निकाय को आव्यूह रूप में लिखने पर: \[ \begin{bmatrix} 2 & 3 & 4 & 1 & | & 3 \\ 1 & 2 & 0 & 1 & | & 2 \\ 2 & 3 & 1 & -1 & | & 1 \\ 1 & -2 & -1 & 4 & | & 5 \end{bmatrix} \] ### चरण 2: पहले स्तंभ में सबसे बड़ा मान वाला पिवट चुनें पहले स्तंभ में सबसे बड़ा मान पहली पंक्ति में है (2)। इसलिए, पहली पंक्ति को पिवट के रूप में चुनें। ### चरण 3: पहली पंक्ति को पिवट के रूप में उपयोग करके अन्य पंक्तियों को शून्य करें पहली पंक्ति को 2 से विभाजित करें: \[ \begin{bmatrix} 1 & 1.5 & 2 & 0.5 & | & 1.5 \\ 1 & 2 & 0 & 1 & | & 2 \\ 2 & 3 & 1 & -1 & | & 1 \\ 1 & -2 & -1 & 4 & | & 5 \end{bmatrix} \] अब, दूसरी पंक्ति से पहली पंक्ति को घटाएं: \[ R_2 = R_2 - R_1 \] \[ \begin{bmatrix} 1 & 1.5 & 2 & 0.5 & | & 1.5 \\ 0 & 0.5 & -2 & 0.5 & | & 0.5 \\ 2 & 3 & 1 & -1 & | & 1 \\ 1 & -2 & -1 & 4 & | & 5 \end{bmatrix} \] तीसरी पंक्ति से पहली पंक्ति को 2 से गुणा करके घटाएं: \[ R_3 = R_3 - 2R_1 \] \[ \begin{bmatrix} 1 & 1.5 & 2 & 0.5 & | & 1.5 \\ 0 & 0.5 & -2 & 0.5 & | & 0.5 \\ 0 ______ _____ _______ _____ ___.
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