GATE CS 2026 Set 1 Q3 — Eigenvalues and Eigenvectors, LU Decomposition

The characteristic equation of an $n \times n$ matrix $A$ is given by:

$$\det(A - \lambda I) = 0$$

Since $A$ is of size $n \times n$, its characteristic polynomial is of degree $n$. By the Fundamental Theorem of Algebra, a polynomial of degree $n$ has exactly $n$ roots (including multiplicities).

The algebraic multiplicity of an eigenvalue is the number of times it appears as a root of this characteristic polynomial. Consequently, the maximum possible multiplicity of any single eigenvalue is $n$, which occurs when all $n$ roots of the characteristic polynomial are identical.

Example:

Consider the $n \times n$ identity matrix $I_n$. Its characteristic equation is:

$$\det(I_n - \lambda I_n) = (1 - \lambda)^n = 0$$

The eigenvalue $\lambda = 1$ has a multiplicity of $n$.

Thus, the maximum multiplicity of any eigenvalue of an $n \times n$ matrix is $n$.

Hence, the correct option is A.