Suppose A is a non-singular matrix such that $A^{3} - 3 A^{2} + 6 A - I = 0. T h e n, A^{- 1}$ =___

$A^{3} - 3 A^{2} + 6 A$

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$A^{2} - 3 A + 6 I$

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$A^{2} + 3 A - 6 I$

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$A^{3} + 3 A^{2} + 6 A$

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Solution

The correct option is B
$A^{2} - 3 A + 6 I$

$A^{3} - 3 A^{2} + 6 A - I = 0 \Rightarrow A^{3} - 3 A^{2} + 6 A = I$
Pre-multiply the equation by $A^{- 1}$
$A^{- 1} A^{3} - 3 A^{- 1} A^{2} + 6 A^{- 1} A = A^{- 1} \Rightarrow (A^{- 1} A) A^{2} - 3 (A^{- 1} A) A + 6 (A^{- 1} A) = A^{- 1} \Rightarrow A^{- 1} = A^{2} - 3 A + 6 I [∵ A^{- 1} A = I]$

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