Question

Consider the 5 x 5 matrix $A=\left[\begin{array}{ccccc}1& 2& 3& 4& 5\\ 5& 1& 2& 3& 4\\ 4& 5& 1& 2& 3\\ 3& 4& 5& 1& 2\\ 2& 3& 4& 5& 1\end{array}\right]$

It is given that A has only one real eigen value. Then the real eigen value A is

• A. -25
• B. 0
• C. 15
• D. 25

Answer 1

The correct option is C 15

Characteristic equation is $|A-\lambda I|$ = 0

$\left|\begin{array}{ccccc}1-\lambda & 2& 3& 4& 5\\ 5& 1-\lambda & 2& 3& 4\\ 4& 5& 1-\lambda & 2& 3\\ 3& 4& 5& 1-\lambda & 2\\ 2& 3& 4& 5& 1-\lambda \end{array}\right|=0$

Using $C_1\to C_1+C_2+C_3+C_4+C_5$

$\left|\begin{array}{ccccc}15-\lambda & 2& 3& 4& 5\\ 15-\lambda & (1-\lambda )& 2& 3& 4\\ 15-\lambda & 5& (1-\lambda )& 2& 3\\ 15-\lambda & 4& 5& (1-\lambda )& 2\\ 15-\lambda & 3& 4& 5& (1-\lambda )\end{array}\right|=0$

$(15-\lambda )\left|\begin{array}{ccccc}1& 2& 3& 4& 5\\ 1& (1-\lambda )& 2& 3& 4\\ 1& 5& (1-\lambda )& 2& 3\\ 1& 4& 5& (1-\lambda )& 2\\ 1& 3& 4& 5& (1-\lambda )\end{array}\right|=0$

So only $\lambda$ = 15 is the real eigen value.