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Which of the following statements about the eigen values of $I_n$, the $n \times n$ identity matrix (over complex numbers), is true?

  1. The eigen values are $1, \omega, \omega^2, \dots , \omega^{n-1}$, where $\omega$ is a primitive $n$-th root of unity
  2. The only eigen value is $-1$
  3. Both 0 and 1 are eigen values, but there are no other eigen values
  4. The eigen values are 1$, 1/2, 1/3, \dots , 1/n$
  5. The only eigen value is 1
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1 Answer

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Best answer

Be it real matrix or complex matrix, the identity matrix remains the same. So we will have in $I_n,$ $1's$ only in the principal diagonal elements and the rest of the elements of the matrix will be $0.$ So the characteristic equation will be :

$\mid A - λ I \mid  =  0$

$\implies \underbrace{(1 - λ) . (1 - λ) \ldots (1 - λ)}_{n \text{ times}}  =   0 $

$\implies (1 - λ)^n    =   0$

$\implies λ  =   1$ as the only solution the reason being complex roots of unity only holds if it were $λ^n  -  1 = 0$ which is not the case here.

Hence (E) is the correct answer.

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