JEE MAIN - Mathematics (2005 - No. 15)
If $${A^2} - A + 1 = 0$$, then the inverse of $$A$$ is :
$$A+I$$
$$A$$
$$A-I$$
$$I-A$$
설명
Given $${A^2} - A + I = 0$$
$${A^{ - 1}}{A^2} - {A^{ - 1}}A + {A^{ - 1}}.I = {A^{ - 1}}.0$$
(Multiplying $$\,\,\,{A^{ - 1}}$$ on both sides)
$$ \Rightarrow A - 1 + {A^{ - 1}} = 0$$
or $${A^{ - 1}} = 1 - A$$
$${A^{ - 1}}{A^2} - {A^{ - 1}}A + {A^{ - 1}}.I = {A^{ - 1}}.0$$
(Multiplying $$\,\,\,{A^{ - 1}}$$ on both sides)
$$ \Rightarrow A - 1 + {A^{ - 1}} = 0$$
or $${A^{ - 1}} = 1 - A$$