##### Which of the above statements about eigenvalues of A is/are necessarily CORRECT?

Let A be n×n real valued square symmetric matrix of rank 2 with . Consider the following statements.

(I) One eigenvalue must be in [−5,5]

(II) The eigenvalue with the largest magnitude must be strictly greater than 5

Which of the above statements about eigenvalues of A is/are necessarily CORRECT?

(A) Both (I) and (II)

(B) (I) only

(C) (II) only

(D) Neither (I) nor (II)

Few words in the question are repeating. Plz correct.

corrected .. now plz solve..

Note two things in the question:

1. Matrix is symmetric.

2. Rank of matrix is 2.

Matrix is of order nxn so n-2 eigen values will be zero.

So eigen values will be, e1, e2, 0, 0, 0......

We have, . = Trace of (A . A

^{T}) = Trace of (A^{2}) = e_{1}^{2}+ e_{2}^{2}+0+0......Hence e

_{1}^{2}+ e_{2}^{2}= 50.So with the above equation, it can be determined that

one eigenvalue must be in [−5,5].Hence B is the correct option.

Matrix is of order nxn so n-2 eigen values will be zero.

So eigen values will be, e1, e2, 0, 0, 0......

how??????

@shweta1920 Matrix is of order nxn so we will get auxiliary equation of order n and roots of this equation will be eigen value. Here it is given that there are only two eigen values, so you can say that all the other roots(eigen values) will be zero.