Q44 / 160 · AP EAMCET 2022 (07 Jul Shift 2) · Mathematics

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

  • A
    $\frac{1}{4}$
  • B
    $41$
  • C
    $31$
  • Correct answer
    D
    $\frac{1}{3} /$

Solution

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

Hence, the correct answer is option: 'D'.

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