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Let $M_{n}(R)$ be the set of n x n matrices with real entries. Which of the following statements is true?

1. Any matrix $A \in M_{4}(R)$ has a real eigenvalue
2. Any matrix $A \in M_{5}(R)$ has a real eigenvalue
3. Any matrix $A \in M_{2}(R)$ has a real eigenvalue
4. None of the above

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.

Option B is correct.

M5 means matrix of order 5, that means polynomial characteristic equation of matrix A is of order 5. So it will give atleast one real root.

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

take this simple example A be a 2x2 matrix with a11 = 3, a12 = -2, a21 = 4 and a22 = -1,

now solve it you will get characteristic equation λ^2 - 2λ + 5 = 0,  so λ = 1 + 2i,  1 - 2i which is complex

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