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Question Number 141193 by Eric002 last updated on 16/May/21

$$A=\left[\begin{array}{ccc}2& 1& -1\\ -3& -1& 2\\ -2& 1& 2\end{array}\right]findtheinverseofthismatrix$$$$$$

Answered by bramlexs22 last updated on 16/May/21

$$Cayley-Hamiltontheorem$$$$\mid A-\lambda I\mid =0$$$$\Rightarrow {\lambda }^3-\left(trA\right){\lambda }^2+\left(\begin{array}{c}minoroftheterms\\ ontheleadingdiagonalofA\end{array}\right)\lambda -det\left(A\right)=0$$$$\Rightarrow {\lambda }^3-3{\lambda }^2-\lambda -1=0$$$$\Rightarrow p\left(A\right)=0$$$$\Rightarrow [A^3-3A^2-A-I=0]\times A^{-1}$$$$\Rightarrow A^2-3A-I-A^{-1}=0$$$$\Rightarrow A^{-1}=A^2-3A-I$$$$$$

Commented by Eric002 last updated on 16/May/21

$$welldonesir$$

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