Consider a 2 × 2 square matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} \sigma &{\text{x}} \\ \omega &\sigma \end{array}} \right]\] where x is unknown. If the eigen values of the matrix A are \[\left( {\sigma + {\text{j}}\omega } \right)\] and \[\left( {\sigma – {\text{j}}\omega } \right)\] , then x is equal to A. \[ + {\text{j}}\omega \] B. \[ – {\text{j}}\omega \] C. \[ + \omega \] D. \[ – \omega \]

The correct answer is $\boxed{\text{B}}$, $-\text{j}\omega$.

The eigenvalues of a matrix are the roots of its characteristic polynomial. The characteristic polynomial of a 2×2 matrix is given by

$$p(x) = |A – xI| = \begin{vmatrix} \sigma & x \\ \omega & \sigma \end{vmatrix} = \sigma^2 – \omega^2$$

The eigenvalues of a matrix are always real numbers. Therefore, the characteristic polynomial of a matrix must have real coefficients. The only way for the characteristic polynomial of a 2×2 matrix to have real coefficients is if $x$ is real. Therefore, $x = -\text{j}\omega$.

Here is a more detailed explanation of each option:

• Option A: $+{\text{j}}\omega$. This is not possible because the eigenvalues of a matrix are always real numbers.
• Option B: $-{\text{j}}\omega$. This is the correct answer.
• Option C: $+\omega$. This is not possible because the eigenvalues of a matrix are always real numbers.
• Option D: $-\omega$. This is not possible because the eigenvalues of a matrix are always real numbers.