× Close
Join the Ques10 Community
Ques10 is a community of thousands of students, teachers, and academic experts, just like you.
Join them; it only takes a minute
Sign up
Question: Are the following matrices derogatory? If yes, find its minimal polynomial.
0
  1. $A=$ $ \left[ {\begin{array}{cc} 6 & -2 & 2\\ -2 & 3 & -1 \\ 0 & 0 & -2\\ \end{array} } \right] $

  2. $A=$ $ \left[ {\begin{array}{cc} 8 & -8 & -2\\ 4 & -3 & -2 \\ 3 & -4 & 1\\ \end{array} } \right] $

Subject: Applied Mathematics 4

Topic: Matrices

Difficulty: Medium

m4m(64) • 62 views
ADD COMMENTlink
modified 18 days ago  • written 18 days ago by gravatar for manasahegde234 manasahegde23410
0
  1. Characteristic equation of A is $\mid A-\lambda I \mid=0$

$\lambda^3-(12)\lambda^2+36\lambda-32=0$

$\lambda=8,2,2.$

$(\lambda-8)(\lambda-2)(\lambda-2)=0$

Consider $(\lambda-8)(\lambda)=\lambda^2-0\lambda+16$

Now to check that $\lambda^2-10A+16I$

Therefore, consider $A^2-10A+16I$

$= \left[ {\begin{array}{cc} 44 & -20 & 20\\ 20 & 14 & -10 \\ 20 & -10 & 14\\ \end{array} } \right] $-$10\left[ {\begin{array}{cc} 6 & -2 & 2\\ -2 & 3 & -1 \\ 2 & -1 & 3\\ \end{array} } \right] $+$16 \left[ {\begin{array}{cc} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1\\ \end{array} } \right] $

$= \left[ {\begin{array}{cc} 44 & -20 & 20\\ 20 & 14 & -10 \\ 20 & -10 & 14\\ \end{array} } \right] $-$10\left[ {\begin{array}{cc} 60 & -20 & 20\\ -20 & 30 & -10 \\ 20 & -10 & 30\\ \end{array} } \right] $+$16 \left[ {\begin{array}{cc} 16 & 0 & 0\\ 0 & 16 & 0 \\ 0 & 0 & 16\\ \end{array} } \right] $

=$16 \left[ {\begin{array}{cc} 0 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0\\ \end{array} } \right] $

$=\lambda^2-10\lambda+16$ annihilates A

Minimal polynomial is $\lambda^2-10\lambda+16$

A is derogatory.

  1. Characteristic equation of A is $\mid A-\lambda I \mid=0$

$\lambda^3-(6)\lambda^2+11\lambda-6=0$

$\lambda=1,2,3$

We have $(\lambda-1)(\lambda-2)(\lambda-3)=0$

All eigen values of A are distinct.

Minimal polynomial of $A=(\lambda-1)(\lambda-2)(\lambda-3)$=$\lambda^3-6\lambda^2+11\lambda-6$= characteristic polynomial.

A is non derogatory.

ADD COMMENTlink
written 18 days ago by gravatar for manasahegde234 manasahegde23410
Please log in to add an answer.


Use of this site constitutes acceptance of our User Agreement and Privacy Policy.