Correct : c
The characteristic polynomial of an n × n matrix is a degree-n polynomial.
det(A − λI) = 0
A degree-n polynomial has exactly n roots (counting multiplicity) over the complex numbers. Therefore, the sum of multiplicities of all eigenvalues can be at most n.
The maximum multiplicity of a single eigenvalue occurs when all n roots of the characteristic polynomial are the same value.
Example: The identity matrix I of size n × n has the characteristic polynomial:
det(I − λI) = (1 − λ)n = 0
This gives λ = 1 with multiplicity n.
Eliminating wrong options:
n + 1 - impossible since the characteristic polynomial is only degree n, it cannot have n+1 roots.
1 - too restrictive, eigenvalues can repeat.
n − 1 - incorrect, as shown by the identity matrix example where multiplicity reaches n.
The maximum multiplicity of any eigenvalue of an n × n real matrix is n (Option C)
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