Question

Give an example of a 2x2 matrix without any real eigenvalues:___________

Answer 1

Explanation:

An eigenvalue of n × n is a function of a scalar
`\lambda` considering that there is a solution (i.e. nontrivial) to an eigenvector x of Ax =

Suppose the matrix
`A = \left[\begin{array}{cc}-1&-1\\2&1\\ \end{array}\right]`

Thus, the equation of the determinant (A -
`\lambda`1) = 0

This implies that:

`\left[\begin{array}{cc}-1-\lambda &-1\\2&1- \lambda\\ \end{array}\right] =0`

`-(1 - \lambda^2 ) + 2 = 0`

`-1 + \lambda ^2 + 2= 0`

`\lambda^2 +1 =0`

Hence, the eigenvalues of the equation are
`\mathtt{\lambda = i , -i}`

Also, the eigenvalues can be said to be complex numbers.