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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2005 • Question 10

Question

Math 6310 • Algebra Fall 2005 • Question 10

Suppose \(F\) is a field and \(T\) is an invertible \(n\times n\) matrix with entries in \(F\). Show that there exist constants \(a_1,\ldots ,a_k\) in \(F\) so that \[ T^{-1}=\sum _{i=1}^k a_iT^i. \]

Raw LaTeX

Suppose \(F\) is a field and \(T\) is an invertible \(n\times n\) matrix with entries in \(F\).
Show that there exist constants \(a_1,\ldots,a_k\) in \(F\) so that
\[
T^{-1}=\sum_{i=1}^k a_iT^i.
\]

Topic Tags

Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization