General Diagonalization Example 2

Question

Determine whether matrix is diagonizable

$$\left[\begin{array}{ccc}7& -4& 0\\ 8& -5& 0\\ 6& -6& 3\end{array}\right]$$

Soln

1. We calculate $X(\lambda )=(\lambda -3{)}^2(\lambda +1)$
2. The geometric multiplicity of $Geo(3)$

4 & -4 & 0 & 0\\ 0 & -8 & 0 & 0\\ 6 & -6 & 0 & 0\\ \end{array}\right]

$Geo(3)=2$ by defn of Rank 3. The geometric multiplicity of $Geo(3)$

8 & -4 & 0 & 0\\ 8 & -4 & 0 & 0\\ 6 & -6 & 4 & 0\\ \end{array}\right] \xrightarrow{REF} \left[\begin{array}{cc} 1 & 0 & -\frac{2}{3} \\ 0 & 1 & -\frac{4}{3}\\ 0 & 0 & 0\\ \end{array}\right]

4. This, $Geo(-1)=Alg(-1)$
5. This, $Geo(3)=Alg31)$
6. The final matrix $M$ is a. We use Diagonalizable definition.