If A is an invertible matrix, such that \(\mathrm{A}^2 - \mathrm{A} + \mathrm{I} = 0\), then the inverse of A is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\mathrm{I} - \mathrm{A}\), where I is an identity matrix of order 2
Here 0 on the right side is the zero matrix O. Since A is invertible, multiply both sides by \(\mathrm{A}^{-1}\):
\(\mathrm{A}^{-1}\mathrm{A}^2 - \mathrm{A}^{-1}\mathrm{A} + \mathrm{A}^{-1}\mathrm{I} = \mathrm{O}\)
\(\mathrm{A} - \mathrm{I} + \mathrm{A}^{-1} = \mathrm{O}\)
\(\mathrm{A}^{-1} = \mathrm{I} - \mathrm{A}\)
Check: \(\mathrm{A}(\mathrm{I} - \mathrm{A}) = \mathrm{A} - \mathrm{A}^2 = \mathrm{I}\), using \(\mathrm{A}^2 = \mathrm{A} - \mathrm{I}\) from the given equation.