 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2019/11/c/ii/solution

M

is invertible, multiply

M^{2} = M

on the left by its matrix inverse:

M^{- 1} M^{2} = M^{- 1} M ⟹ M = I .

Thus a nonidentity idempotent matrix must be singular; the identity itself is both invertible and idempotent.

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