There is a real 5x5 matrix with no real eigenvalues Prove or

There is a real 5x5 matrix with no real eigenvalues.
Prove or disprove showing steps clearly.

Solution

Since the matrix has real entries, its characteristic polynomial is of order 5 and has real coefficients.

Since any odd-powered polynomial having real coefficients must intersect the horizontal axis at least once when plotted against y, every real 5x5 matrix must have at least one real eigenvalue.