If A 2 − A + I = 0 , then the inverse of the matrix A is, A − I, I − A, A + I, A

Hints : A 2 − A + I = 0 ⇒ A 2 = A − I ⇒ A 2 ⋅ A − 1 = A ⋅ A − 1 − A − 1 ⇒ A = I − A − 1 ⇒ A − 1 = I − A

If A 2 − A + I = 0 , then the inverse of the matrix A is

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If $A^2-A+I=0$, then the inverse of the matrix $A$ is

MathematicsMatricesWBJEEWBJEE 2009

Options:

A $\mathrm{A}-\mathrm{I}$
B $\quad \mathrm{I}-\mathrm{A}$
C $\mathrm{A}+\mathrm{I}$
D $\mathrm{A}$

Solution:

2905 Upvotes Verified Answer

The correct answer is: $\quad \mathrm{I}-\mathrm{A}$

Hints : $\mathrm{A}^2-\mathrm{A}+\mathrm{I}=0 \Rightarrow \mathrm{A}^2=\mathrm{A}-\mathrm{I} \Rightarrow \mathrm{A}^2 \cdot \mathrm{A}^{-1}=\mathrm{A} \cdot \mathrm{A}^{-1}-\mathrm{A}^{-1} \Rightarrow \mathrm{A}=\mathrm{I}-\mathrm{A}^{-1} \Rightarrow \mathrm{A}^{-1}=\mathrm{I}-\mathrm{A}$

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