In the question M_{n}(R) be the set of n x n matrices with real entries, all matrices are real, how can they give rise to complex numbers?

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Option $b$.

The eigenvalues are found by solving the characteristic equation. For a matrix $M_n$, the degree of the characteristic polynomial/equation will be $n$ and hence, there will be exactly $n$ roots. These roots can be real or complex. But we know that for an equation with real coefficients, complex roots (*if any*) will exist in conjugate pairs. So, for an odd value of $n$, even if there exist complex roots, there will be at least one real root.