यदि A=left[begin{array}{ccc}2 & -1 & 1 -1 & | vedclass

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(A) दिया गया है $A=\left[\begin{array}{ccc}2 & -1 & 1 \ -1 & 2 & -1 \ 1 & -1 & 2\end{array}ight]$.सबसे पहले,$A^{2} = A 	imes A = \left[\begin{arr

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$\frac{1}{4}\left[\begin{array}{ccc}3 & 1 & -1 \ 1 & 3 & 1 \ -1 & 1 & 3\end{array}ight]$

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(A) दिया गया है $A=\left[\begin{array}{ccc}2 & -1 & 1 \ -1 & 2 & -1 \ 1 & -1 & 2\end{array}ight]$.सबसे पहले,$A^{2} = A 	imes A = \left[\begin{array}{ccc}2 & -1 & 1 \ -1 & 2 & -1 \ 1 & -1 & 2\end{array}ight] \left[\begin{array}{ccc}2 & -1 & 1 \ -1 & 2 & -1 \ 1 & -1 & 2\end{array}ight] = \left[\begin{array}{ccc}6 & -5 & 5 \ -5 & 6 & -5 \ 5 & -5 & 6\end{array}ight]$ ज्ञात करें।इसके बाद,$A^{3} = A^{2} 	imes A = \left[\begin{array}{ccc}6 & -5 & 5 \ -5 & 6 & -5 \ 5 & -5 & 6\end{array}ight] \left[\begin{array}{ccc}2 & -1 & 1 \ -1 & 2 & -1 \ 1 & -1 & 2\end{array}ight] = \left[\begin{array}{ccc}22 & -21 & 21 \ -21 & 22 & -21 \ 21 & -21 & 22\end{array}ight]$ ज्ञात करें।अब,$A^{3}-6 A^{2}+9 A-4 I = \left[\begin{array}{ccc}22 & -21 & 21 \ -21 & 22 & -21 \ 21 & -21 & 22\end{array}ight] - 6\left[\begin{array}{ccc}6 & -5 & 5 \ -5 & 6 & -5 \ 5 & -5 & 6\end{array}ight] + 9\left[\begin{array}{ccc}2 & -1 & 1 \ -1 & 2 & -1 \ 1 & -1 & 2\end{array}ight] - 4\left[\begin{array}{ccc}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}ight] = \left[\begin{array}{ccc}0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0\end{array}ight] = 0$ सत्यापित करें।$A^{-1}$ ज्ञात करने के लिए,समीकरण $A^{3}-6 A^{2}+9 A-4 I=0$ को $A^{-1}$ से गुणा करें:$A^{2}-6 A+9 I-4 A^{-1}=0 \Rightarrow 4 A^{-1} = A^{2}-6 A+9 I$.$4 A^{-1} = \left[\begin{array}{ccc}6 & -5 & 5 \ -5 & 6 & -5 \ 5 & -5 & 6\end{array}ight] - 6\left[\begin{array}{ccc}2 & -1 & 1 \ -1 & 2 & -1 \ 1 & -1 & 2\end{array}ight] + 9\left[\begin{array}{ccc}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}ight] = \left[\begin{array}{ccc}3 & 1 & -1 \ 1 & 3 & 1 \ -1 & 1 & 3\end{array}ight]$.अतः,$A^{-1} = \frac{1}{4}\left[\begin{array}{ccc}3 & 1 & -1 \ 1 & 3 & 1 \ -1 & 1 & 3\end{array}ight]$.

यदि $A=\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2\end{array}\right]$ है,तो सत्यापित कीजिए कि $A^{3}-6 A^{2}+9 A-4 I=0$ और इसके प्रयोग से $A^{-1}$ ज्ञात कीजिए।

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