Question

Let.

1             0             0

0             3             6

0             -1           -2

Verify that C ² = C holds. Find the eigenvalues and eigenvectors of C. 

Answer 1

$C^2=\begin{pmatrix} 1 & 0&0 \\ 0 & 3&6\\ 0&-1&-2 \end{pmatrix}\begin{pmatrix} 1 & 0&0 \\ 0 & 3&6\\ 0&-1&-2 \end{pmatrix}=\begin{pmatrix} 1 & 0&0 \\ 0 & 10&22\\ 0&-2&-2 \end{pmatrix}$

$\begin{vmatrix} 1-\lambda & 0&0 \\ 0&3- \lambda& 6\\ 0&-1&-2-\lambda \end{vmatrix}=0$

$(1-\lambda)((3-\lambda)(-2-\lambda)+6)=0$

$\lambda_1=1$

$\lambda^2-\lambda=0$

$\lambda_2=0,\lambda_3=1$

for $\lambda=1$ :

$2y+6z=0$

$-y-3z=0$

$y=-3z$

$u_1=\begin{pmatrix} 1 \\ -1\\ 3 \end{pmatrix},u_2=\begin{pmatrix} 1 \\ 1\\ -3 \end{pmatrix}$

for $\lambda=0$ :

$x=0$

$3y+6z=0$

$-y-2z=0$

$y=-2z$

$u_3=\begin{pmatrix} 0 \\ -1\\ 2 \end{pmatrix}$