# Eigenvalues and eigenvectors of a given matrix $A$

I have this matrix

$$A = \left( {\begin{array}{ccc} 1 & 1 & -1 \\ 0 & 2 & 0 \\ 0 & 1 & -1 \end{array} } \right)$$

and I have to find eigenvalues and eigenvectors of $A$

I am new to this can anyone help?

• Work out a few examples here Oct 25 '15 at 18:37
• Note that $A$ has an upper triangular block structure. This means that you can split the problem into solving the upper left block (which is just 1) and solving the lower right block and combining the answers. Oct 25 '15 at 18:43
• as Weaam comment, it is worth to use another resources wolframalpha.com/input/… Oct 25 '15 at 18:46

You can find the roots of the characteristic polynomial $$\chi_A(X):=\det(A-X\mathrm{I_3}).$$ Indeed, you know that $\lambda$ is an eigenvalue if it exists $x\in\mathbb{R}^3-\{(0,0,0)\}$ as $AX=\lambda X$. It means that $$(A-\lambda\mathrm{I_3})(X)=0,$$ and thus that $A-\lambda\mathrm{I_3}$ is not regular. This happens if and only if $\det(A-\lambda\mathrm{I_3})=0$, which give you a first way to answer your problem. Once you have found $\lambda$ an eigenvalue, you get an eigenvector (associated to $\lambda$) by finding $x\in\ker(A-\lambda\mathrm{I}_3)$.