# A*a + i = a

#### Also sprach Zarathustra

Prove that there not such matrix A in M_(nxn)(C) so that:
A*A + I = A.

(When A The conjugate transpose of the complex matrix A)

Thanks!

#### Opalg

MHF Hall of Honor
Prove that there not such matrix A in M_(nxn)(C) so that:
A*A + I = A.

(When A The conjugate transpose of the complex matrix A)
Notice that $$\displaystyle A^* = (A^*A+I)^* = A^*A+I = A$$. Therefore $$\displaystyle A^2-A+I=0$$ and it follows that each eigenvalue of A must satisfy the equation $$\displaystyle \lambda^2-\lambda+1=0$$. But that equation does not have real roots, which contradicts the fact that $$\displaystyle A^*=A$$.

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