Solution
A=( egin{pmatrix}2&0&0&0\ 0&2&0&0\ 0&0&1&0\ 0&0&0&3end{pmatrix} )
The characteristics polynomial. ( implies P(lambda)=|A-lambda I|=igg(2-lambdaigg)^2igg(1-lambdaigg)igg(3-lambdaigg)hspace{2mm} )
The algebraic multiplicity
( implies am(2)=2,am(1)=1,am(3)=1 hspace{2mm} )
The geometric multiplicity
( implies gm(2)=2,gm(1)=1,gm(3)=1hspace{2mm} )
The ( indeximplies Index(2)=1,Index(1)=1,Index(3)=1 hspace{2mm} )
The minimal polynomial
( implies m(lambda)=igg(lambda-1igg)igg(lambda-1igg)igg(lambda-3igg)hspace{2mm} )
Answer :
Therefore .
( implies P(lambda)=|A-lambda I|=igg(2-lambdaigg)^2igg(1-lambdaigg)igg(3-lambdaigg)hspace{2mm} )
( implies am(2)=2,am(1)=1,am(3)=1 hspace{2mm} )
( implies gm(2)=2,gm(1)=1,gm(3)=1hspace{2mm} )
( implies Index(2)=1,Index(1)=1,Index(3)=1 hspace{2mm} )
