Word Count : 443

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आव्यूह \( A = \begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix} \) के लिए कैली-हैल्टन प्रमेय (Cayley-Hamilton Theorem) सत्यापित करने के लिए, हमें सबसे पहले \( A \) का विशेष गुणांक (characteristic polynomial) निकालना होगा और फिर यह प्रमेय लागू करना होगा। 1. विशेष गुणांक निकालना: विशेष गुणांक को निकालने के लिए, हम \( \lambda \) के लिए गुणांक समीकरण \( \text{det}(A - \lambda I) = 0 \) हल करेंगे, जहाँ \( I \) एक अभिन्न समीकरण (identity matrix) है। यहाँ, \( A - \lambda I = \begin{bmatrix} 1 - \lambda & -1 \\ 0 & 2 - \lambda \end{bmatrix} \) है। अब, गुणांक समीकरण: \[ \text{det}(A - _________ __________ ___ __________ __________ ________ ________.
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