Question

For \(n > 1\), the maximum multiplicity of any eigenvalue of an \(n \times n\) matrix with elements from \(\mathbb{R}\) is

• A. \(n\)
• B. \(n - 1\)
• C. \(1\)
• D. \(n + 1\)

Hint:

The characteristic polynomial of an \(n \times n\) matrix has degree \(n\), and a scalar matrix like \(cI_n\) gives one eigenvalue with multiplicity exactly \(n\).

Answer 1

The Correct Option is A

Step 1: Concept used. For a square matrix \(A\) of order \(n\), the eigenvalues are the roots of the characteristic equation \(\det(A - \lambda I) = 0\). This characteristic polynomial in \(\lambda\) always has degree exactly \(n\). The algebraic multiplicity of a particular eigenvalue is the number of times that value repeats as a root of this degree \(n\) polynomial.

Step 2: Upper bound on multiplicity. Since the characteristic polynomial has degree \(n\), it can have at most \(n\) roots counted with multiplicity. So no single eigenvalue of an \(n \times n\) matrix can have algebraic multiplicity greater than \(n\). This rules out option (D), \(n + 1\), because a multiplicity that large is not possible for any \(n \times n\) matrix.

Step 3: Check whether the bound \(n\) is actually attainable. Take the scalar matrix \(A = cI_n\), where \(c\) is any real number and \(I_n\) is the \(n \times n\) identity matrix. All entries of \(A\) are real, so this is a valid matrix with elements from \(\mathbb{R}\). Its characteristic polynomial is \(\det(cI_n - \lambda I_n) = \det((c-\lambda)I_n) = (c-\lambda)^n\). This has the single root \(\lambda = c\), repeated \(n\) times, so the eigenvalue \(c\) has algebraic multiplicity exactly \(n\) for this matrix.

Step 4: Conclusion. Multiplicity \(n\) is both an upper bound (Step 2) and an achievable value (Step 3), so the maximum possible multiplicity of any eigenvalue of an \(n \times n\) real matrix is exactly \(n\).

Step 5: Why the other options are wrong. Option (B), \(n-1\), is smaller than the true maximum \(n\), which has already been shown achievable by \(cI_n\). Option (C), \(1\), is actually the smallest possible value the maximum multiplicity can be forced down to, namely when all \(n\) eigenvalues are distinct, for example a diagonal matrix with \(n\) different real diagonal entries, it is not the maximum over all matrices. Option (D), \(n+1\), is impossible because the characteristic polynomial has degree only \(n\), so no root can be repeated \(n+1\) times.

\[ \boxed{n} \]