If a matrix $A$ satisfies the equation $A^3-6 A^2+11 A-6 I=0$, then $A^{-1}$ can be

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Aicharya · Accepted Answer

$ \begin{aligned} & A^3-6 A^2+11 A-6 I=0 \ & A^3-A^2-5 A^2+5 A+6 A-6 I=0 \ & \Rightarrow A^2[A-I]-5 A[A-I]+6[A-I]=0 \ & \Rightarrow {[A-I]\left[A^2-5 A+6ight]=0 } \ & \Rightarrow(A-I)(A-2 I)(A-3 I)=0 \end{aligned} $ $	herefore A-3 I$ can be a zero matrix. $ \begin{aligned} \Rightarrow & & A & =3 I \ \Rightarrow & & A A^{-1} & =3 A^{-1} \ \Rightarrow & & I & =3 A^{-1} \ \Rightarrow & & A A^{-1} & =\frac{1}{3} I \end{aligned} $

If a matrix $A$ satisfies the equation $A^3-6 A^2+11 A-6 I=0$, then $A^{-1}$ can be

Solution