This question was previously asked in

GATE PI 2016 Official Paper

Option 1 : i and –i

__Concept:__

If A is any square matrix of order n, we can form the matrix [A – λI], where I is the nth order unit matrix. The determinant of this matrix equated to zero i.e. |A – λI| = 0 is called the characteristic equation of A.

The roots of the characteristic equation are called** Eigenvalues or latent roots or characteristic roots of matrix A.**

Eigenvector (X) that corresponding to Eigenvalue (λ) satisfies the equation AX = λX.

Properties of Eigenvalues:

The sum of Eigenvalues of a matrix A is equal to the trace of that matrix A

The product of Eigenvalues of a matrix A is equal to the determinant of that matrix A

__Calculation:__

__Given:__

A = \(\begin{bmatrix} 0&1 \\ -1 &0 \end{bmatrix}\)

For characteristic equation:

|A – λI| = 0

\(\begin{vmatrix} 0 - \lambda &1 \\ -1 & 0 - \lambda \end{vmatrix}\) = 0

\(\lambda \)^{2} + 1 = 0

\(\lambda \) = i, -i