If A2- A +I=0, then the inverse of A is

Question

If A2- A +I=0, then the inverse of A is (A) I-A (B) A-I (C) A (D) A+I

Tardigrade Admin · Accepted Answer

If A is any square matrix, then AA1 = I and A1 I = A-1 Since, A2 - A + 1 = 0 ⇒ A-1 A2 - A-1 A + A1 I = O ⇒ IA - I + A-1 = 0 ⇒ A - I + A-1 = 0 ⇒ A-1 = I - A

Q. If $A^{2} - A + I = 0$ , then the inverse of $A$ is

$I - A$

$A - I$

$A$

$A + I$

If $A$ is any square matrix, then

$A A^{1} = I$ and $A^{1} I = A^{- 1}$

Since, $A^{2} - A + 1 = 0$

$\Rightarrow A^{- 1} A^{2} - A^{- 1} A + A^{1} I = O$

$\Rightarrow I A - I + A^{- 1} = 0$

$\Rightarrow A - I + A^{- 1} = 0$

$\Rightarrow A^{- 1} = I - A$

Q. If A2−A+I=0, then the inverse of A is

I−A

A−I

A

A+I

If A is any square matrix, then AA1=I and A1I=A−1 Since, A2−A+1=0 ⇒A−1A2−A−1A+A1I=O ⇒IA−I+A−1=0 ⇒A−I+A−1=0 ⇒A−1=I−A

Q. If $A^{2} - A + I = 0$ , then the inverse of $A$ is

$I - A$

$A - I$

$A$

$A + I$

If $A$ is any square matrix, then

$A A^{1} = I$ and $A^{1} I = A^{- 1}$

Since, $A^{2} - A + 1 = 0$

$\Rightarrow A^{- 1} A^{2} - A^{- 1} A + A^{1} I = O$

$\Rightarrow I A - I + A^{- 1} = 0$

$\Rightarrow A - I + A^{- 1} = 0$

$\Rightarrow A^{- 1} = I - A$