Science:Math Exam Resources/Courses/MATH307/April 2010/Question 02 (d)/Statement

A real matrix $A_{5\times 5}$ is brought into its Jordan canonical form

$B={\begin{pmatrix}1&1&0&0&0\\0&1&0&0&0\\0&0&0&0&0\\0&0&0&-1&0\\0&0&0&0&-1\end{pmatrix}}$

by a matrix $S_{5\times 5}$, i.e. $\displaystyle A=SBS^{-1}$. Answer the questions below, substantiate all your statements.

(d) What is the determinant of the product matrix $\displaystyle CA$ if $C\in \mathbb {C} ^{5\times 5}$ is a Hermitian matrix with full geometric multiplicity?