If A is a non-singular matrix and \(A^2-A+I=0\), then the inverse of A is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(I-A\)
Here 0 on the right-hand side is the zero matrix O. Since A is non-singular, \(A^{-1}\) exists. Multiplying the given equation by \(A^{-1}\):
\(A^{-1}A^2-A^{-1}A+A^{-1}I=O\)
\(A-I+A^{-1}=O\)
\(A^{-1}=I-A\)
Check: from the given equation \(A-A^2=I\), so \(A(I-A)=A-A^2=I\), which confirms that \(I-A\) is the inverse of A.