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Matrices and Determinants question

2005 · Shift 0 · Q74

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Matrices and Determinants
2005 · Shift 0 · Q74

Matrices and Determinants question

2005 · Shift 0 · Q74

JEE MainMathematicsMatrices and DeterminantsMCQ+4 / −1

If

{A^2} - A + 1 = 0

, then the inverse of

A

is :

A
$A+I$
B
$A$
C
$A-I$
D
$I-A$

View written solutionFree

Correct answer: D

We are given $A^2 - A + I = 0$ where $I$ is the identity matrix.
Rearrange the equation: $A^2 - A = -I$
Factor out $A$ : $A(A - I) = -I$
Multiply both sides by $-1$ : $A(I - A) = I$
Since a matrix $X$ satisfying $AX = I$ is the inverse of $A$ , we get $A^{-1} = I - A$
Therefore, the correct option is: $\boxed{I - A}$
Checking options:
- A: $A+I$ ❌
- B: $A$ ❌
- C: $A-I$ ❌
- D: $I-A$ ✅

Hence, the inverse of $A$ is $I-A$ .

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