Let A be an n × n matrix such that the set of all its nonzero eigenvalues has exactly r elements. Which of the following statements is true?

Calculation: 

Let A be an n × n matrix such that the set of all its nonzero eigenvalues has exactly r elements.

let E = { a₁ , a₂ , . . . . .  a_r} 

for each non zero eigen values there is at least one eigen vector .

for r non zero distinct eigenvector .

range space is at least r .

Hence option 3 is correct .

Option (1): 

Let A = \(\begin{bmatrix}1&0\\0&0\end{bmatrix}\) then eigenvalues are 0, 0 ⇒ r = 0

rank(A) = 1 = 2 - 1 \(\nless \) 2 - 1

Option (2) is false

Rank(A) = 1 \(\nless \) r = 0

Option (1) is false

Option (4):

A has r non-zero eigenvalues

⇒ A² has r non-zero eigenvalues

But if A has r distinct eigenvalues does not imply A2 has r distinct eigenvalues.

Let A = \(\begin{bmatrix}0&-1\\1&0\end{bmatrix}\) then eigenvalues of A are i, -1

but A2 has eigenvalues -1, -1 which are not distinct.

Option (4) is false