# How to find the eigenvalue, eigenspace and minimal polynomial of this matrix? [closed]

Let $A=\begin{pmatrix} -a &a+2 & -3a-4 \\ -a+1 & a+1 & -3a+2 \\ 1 &-1 &4 \end{pmatrix}$. where $a\in \mathbb C$.

Then what are the eigenvalue, eigenspace and minimal polynomial of $A$? I could calculate when $a=0$ and then the eigenvector was $(-2t, t, t)$

## closed as off-topic by Claude Leibovici, Davide Giraudo, ronno, Rafa Budría, B. MehtaJun 3 '17 at 22:47

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## 1 Answer

It's Eigen values are 1,2,2

Simply find the Eigen values as you find for any $3 \times 3$ matrix

Using this formula

Characteristic equation$=x^3-(\text{trace} A)x^2+(\text{trace of adjoint} A)x - \det A$

In the above ques Trace $A=5$, $\det A=4$, trace of adjoint $A=8$

So characteristic equation will be $$x^3-5x^2+8x-4$$

Solving this you get eigen values=1,2,2,

• Please format your answer using $\LaTeX$ to make it more readable. Moving mathematical equations inside dollar signs (\\$) should do the majority of the work. – NoseKnowsAll Jun 3 '17 at 5:58
• I dont know how to do that.. – user450820 Jun 3 '17 at 6:09