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If |A| 0 and (A-2I)(A-3I)=0 then A^{-1}=?

Question

If $∣ A ∣ \neq = 0$ and $(A - 2 I) (A - 3 I) = 0$ then $A^{- 1} = ?$

\frac{A - 5 I}{6}

\frac{5 I - A}{5}

\frac{5 A - I}{6}

\frac{5 I - A}{6}

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Oliver Discussed

∣ A ∣ \neq = 0

and

(A - 2 I) (A - 3 I) = 0

then

A^{- 1} = ?

5 mins ago

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Text solutionVerified

(A - 2 I) (A - 3 I) = 0, A^{- 1} = ? \Rightarrow A \cdot A - 3 A \cdot I - 2 I \cdot A + 6 I \cdot I = O [∵ A \cdot I = A] \Rightarrow A \cdot A - 3 A - 2 A + 6 I = 0 \Rightarrow A \cdot A + 6 I = 5 A

Multiply

A^{- 1}

on both side

\Rightarrow A \cdot A (A^{- 1}) + 6 I (A^{- 1}) = 5 A (A^{- 1}) \Rightarrow A \cdot (A A^{- 1}) + 6 A^{- 1} = 5 I [∵ A A^{- 1} = I & ∵ I A^{- 1} = A^{- 1}] \Rightarrow A I + 6 A^{- 1} = 5 I \Rightarrow A^{- 1} = \frac{5 I - A I}{6} \Rightarrow A^{- 1} = \frac{5 I - A}{6}

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