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## Homework Statement

If A is an invertible idempotent matrix, then A must be the Identity matrix I_n.

## Homework Equations

A^2==A ; A^2==AA; A^(-1); I==A^(-1)

## The Attempt at a Solution

Suppose A is an nxn matrix =/= I_n.

s.t. A^(2)==A

so A^(2)==A ==> AA==A

==> A^(-1)AA==A^(-1)A ==> A==I==> A^(-1)A==A^(-1)I==>I==A^(-1)I==A^(-1)==A

which yeilds a contradiction because we supposed our A =/= I_n.

Therefore A==I_n

Is this correct please help me understand where I have failed...