Solution Â
we have B=\( \begin{pmatrix}2&1&0&0\\ 0&2&0&0\\ 0&0&3&0\\ 0&0&0&3\end{pmatrix} \)
The characteristics polynomial.
\( \implies P(\lambda)=|B-\lambda I|=\bigg(2-\lambda\bigg)^2\bigg(3-\lambda\bigg)^3 \)
The algebraic multiplicity
\( \implies am(2)=2,am(3)=2 \)
The geometric multiplicity \( \implies gm(2)=1,gm(3)=2 \)
The \( index\implies Index(2)=2,Index(3)=1 \)
The minimal polynomial. \( \implies m(\lambda)=\bigg(\lambda-2\bigg)^2\bigg(\lambda-3\bigg)^3 \)
Answer
Therefore.Â
\( \implies P(\lambda)=|B-\lambda I|=\bigg(2-\lambda\bigg)^2\bigg(3-\lambda\bigg)^3 \)
\( \implies am(2)=2,am(3)=2 \)
\( \implies gm(2)=1,gm(3)=2 \)
\( \implies Index(2)=2,Index(3)=1 \)
\( \implies m(\lambda)=\bigg(\lambda-2\bigg)^2\bigg(\lambda-3\bigg)^3 \)
