Question

Let A be (n x n) real valued square symmetric matrix of rank 2 with \(\mathop \sum \limits_{i = 1}^n \mathop \sum \limits_{j = 1}^n A_{ij}^2 = 50\). 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?
1. Both I and II
2. I only
3. II only
4. Neither I nor II

Answer 1

Correct Answer - Option 2 : I only

Try to find out the eigen values by taking the example of matrix with rank 2. Then check if given statement follows from that or not.

Calculation:

A be (n x n) real valued square symmetric matrix of rank 2.

\(\mathop \sum \limits_{i = 1}^n \mathop \sum \limits_{j = 1}^n A_{ij}^2 = 50\),

which means, sum of square of all elements of A = 50.

Also, rank of A = 2 i.e. we have (n – 2) eigen values are 0.

So eigen values are in the form of a₁, a_2,0, 0, ……

Let us consider a matrix A = \(\left[ {\begin{array}{*{20}{c}} { - 5}&0\\ 0&5 \end{array}} \right]\)

Here, for this eigen values are [-5, 5].

Take another example, B = \(\left[ {\begin{array}{*{20}{c}} 5&0&0\\ 0&5&0\\ 0&0&0 \end{array}} \right]\) and C = \(\left[ {\begin{array}{*{20}{c}} 6&0&0\\ 0&{\surd 14}&0\\ 0&0&0 \end{array}} \right]\)

Matrix B and C are of rank 2 and symmetric.

Eigen value for A = 5, 5, 0

And eigen value for B = \(6,\;\sqrt {14} ,\;0\)

As 0 and 5 are in the range of [-5, 5], So statement 1 is correct.

But eigen value with largest magnitude which is 6 for C must be strictly greater than 5 and in case of B largest value is 5 which is not strictly greater than 5, So, second statement is incorrect here.