Question

The value of x for which all the eigen values of the matrix given below are real is
$\left[\begin{array}{ccc}10& 5+j& 4\\ x& 20& 2\\ 4& 2& -10\end{array}\right]$

• A. 5 + j
• B. 5 - j
• C. 1 - 5j
• D. 1 + 5j

Answer 1

The correct option is B 5 - j

Given that

A = $\left[\begin{array}{ccc}10& 5+j& 4\\ x& 20& 2\\ 4& 2& -10\end{array}\right]$

$\Rightarrow$ $\left[\overline{A}\right]=\left[\begin{array}{ccc}10& 5-j& 4\\ \overline{x}& 20& 2\\ 4& 2& -10\end{array}\right]$

$\Rightarrow A^{\theta }=[\overline{A}{]}^T=\left[\begin{array}{ccc}10& \overline{x}& 4\\ 5-j& 20& 2\\ 4& 2& -10\end{array}\right]$

We know that "Eigen value of a Hermitian matrix are real numbers"

A is Hermitian matrix if $A^{\theta }$ = A

$\therefore \left[\begin{array}{ccc}10& \overline{x}& 4\\ 5-j& 20& 2\\ 4& 2& -10\end{array}\right]=\left[\begin{array}{ccc}10& 5+j& 4\\ x& 20& 2\\ 4& 2& -10\end{array}\right]$

$\Rightarrow \overline{x}=5+jandx=5-j$

$\therefore$ The value of x = 5 - j