Let A be a 3 × 3 real matrix whose characteristic polynomial p(T) is divisible by T². Which of the following statements is true?

Explanation:

Characteristic polynomial p(T) is divisible by T2.

p(x)/x2

So p(x) = x2(x + a) where a can be zero also

Option (1): Let A = \(\begin{bmatrix}0&2&3\\0&0&1\\0&0&a\end{bmatrix}\) Here 0 and a are the eigenvalues and eigenspace of A for 0 is 1

So option (1) is false

Here A is 3 × 3 matrix and two eigenvalues are 0, 0. Since complex eigenvalue are always complex conjugate and they are in pairwise. So here third eigenvalue must be real.

Option (2) is correct

 For A = \(\begin{bmatrix}0&2&3\\0&0&1\\0&0&a\end{bmatrix}\), A3 ≠ 0

Option (3) is false

also AM of eigenvalue 0 is 2 and GM of eigenvalue 0 is 1 

Since AM ≠ GM so not diagonalizable.

Option (4) is false