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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2025 • Question 3

Question

Math 6310 • Algebra Fall 2025 • Question 3

Suppose that \(A\) is an \(n \times n\) matrix with characteristic polynomial

\[ (x-1)^n \]

Prove that if \(A^d=A^e\) for any pair \(d\ne e\) of integers, then \(A=I_n\). And no, you cannot just quote the theorem that a matrix of finite order is diagonalizable. We want you to demonstrate that you know why it is true.

Raw LaTeX

Suppose that \(A\) is an \(n \times n\) matrix with characteristic polynomial

\[
(x-1)^n
\]

Prove that if \(A^d=A^e\) for any pair \(d\ne e\) of integers, then \(A=I_n\).
And no, you cannot just quote the theorem that a matrix of finite order is
diagonalizable. We want you to demonstrate that you know why it is true.

Topic Tags

Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization Polynomial Rings and Ring Homomorphism Kernels