Question

Find the eigenvalues and eigenvectors of A geometrically over the real numbers ℝ. (If an eigenvalue does not exist, enter DNE. If an eigenvector does not exist, enter DNE in any single blank.) A = 0 −1 −1 0 (reflection in the line y = −x)

Answer 1

`v_1=(1,1)\\v_2=(-1,1)`

Explanation:

you have the matrix

`A=\left[\begin{array}{cc}0&-1\\-1&0\end{array}\right]`

to find the eigenvalues it is necessary to solve the determinant

`|A-\lambda I|=0\\`

thus, we have the polynomial and the eigenvalues

`\lambda ^2-1=0\\\lambda_1=1\\\lambda_2=-1`

and the eigenvectors

`v_1=(1,1)\\v_2=(-1,1)`

Hope this helps!