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Dosmascerveza
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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_nIs this correct please help me understand where I have failed...