- #1
VinnyCee
- 489
- 0
Here is the problem:
[tex]X'\,=\,\left(\begin{array}{ccc}3 & -1 & -1\\1 & 1 & -1\\1 & -1 & 1\end{array}\right)\,X[/tex]
Here is what i have so far:
[tex]det(A\,-\,rI)\,=\,0[/tex]
[tex]det\left[\left(\begin{array}{ccc}3 & -1 & -1\\1 & 1 & -1\\1 & -1 & 1\end{array}\right)\,-\,\left(\begin{array}{ccc}r & 0 & 0\\0 & r & 0\\0 & 0 & r\end{array}\right)\right]\,=\,\left(\begin{array}{ccc}3\,-\,r & -1 & -1\\1 & 1\,-\,r & -1\\1 & -1 & 1\,-\,r\end{array}\right)[/tex]
[tex]-(r\,-\,1)\,(r^2\,-\,4r\,+4)\,=\,0[/tex]
[tex]r_1\,=\,1,\,\,\,\,r_2\,=\,2,\,\,\,\,r_3\,=\,2[/tex]
[itex]r\,=\,2[/itex] is repeated once.
[tex](A\,-\,r_1I)\,\xi\,=\,0[/tex]
[tex]\left[\left(\begin{array}{ccc}3 & -1 & -1\\1 & 1 & -1\\1 & -1 & 1\end{array}\right)\,-\,\left(\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array}\right)\right]\,\left(\begin{array}{ccc} \xi_1 \\ \xi_2 \\ \xi_3 \end{array}\right)=\,\left(\begin{array}{ccc}2 & -1 & -1\\1 & 0 & -1\\1 & -1 & 0\end{array}\right)\,\left(\begin{array}{ccc} \xi_1 \\ \xi_2 \\ \xi_3 \end{array}\right)\,=\,0[/tex]
Now, multiplying out the right hand side of the equation above, and rref'ing, I get:
[tex]\left(\begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right)[/tex]
Which means that,
[tex]\xi_1\,-\,\xi_3\,=\,0[/tex]
[tex]\xi_2\,-\,\xi_3\,=\,0[/tex]
So then,
[tex]\xi_1\,=\,\xi_2\,=\,\xi_3?[/tex]
Here is where I am stuck! What am I supposed to do now to solve this system?
[tex]X'\,=\,\left(\begin{array}{ccc}3 & -1 & -1\\1 & 1 & -1\\1 & -1 & 1\end{array}\right)\,X[/tex]
Here is what i have so far:
[tex]det(A\,-\,rI)\,=\,0[/tex]
[tex]det\left[\left(\begin{array}{ccc}3 & -1 & -1\\1 & 1 & -1\\1 & -1 & 1\end{array}\right)\,-\,\left(\begin{array}{ccc}r & 0 & 0\\0 & r & 0\\0 & 0 & r\end{array}\right)\right]\,=\,\left(\begin{array}{ccc}3\,-\,r & -1 & -1\\1 & 1\,-\,r & -1\\1 & -1 & 1\,-\,r\end{array}\right)[/tex]
[tex]-(r\,-\,1)\,(r^2\,-\,4r\,+4)\,=\,0[/tex]
[tex]r_1\,=\,1,\,\,\,\,r_2\,=\,2,\,\,\,\,r_3\,=\,2[/tex]
[itex]r\,=\,2[/itex] is repeated once.
[tex](A\,-\,r_1I)\,\xi\,=\,0[/tex]
[tex]\left[\left(\begin{array}{ccc}3 & -1 & -1\\1 & 1 & -1\\1 & -1 & 1\end{array}\right)\,-\,\left(\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array}\right)\right]\,\left(\begin{array}{ccc} \xi_1 \\ \xi_2 \\ \xi_3 \end{array}\right)=\,\left(\begin{array}{ccc}2 & -1 & -1\\1 & 0 & -1\\1 & -1 & 0\end{array}\right)\,\left(\begin{array}{ccc} \xi_1 \\ \xi_2 \\ \xi_3 \end{array}\right)\,=\,0[/tex]
Now, multiplying out the right hand side of the equation above, and rref'ing, I get:
[tex]\left(\begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right)[/tex]
Which means that,
[tex]\xi_1\,-\,\xi_3\,=\,0[/tex]
[tex]\xi_2\,-\,\xi_3\,=\,0[/tex]
So then,
[tex]\xi_1\,=\,\xi_2\,=\,\xi_3?[/tex]
Here is where I am stuck! What am I supposed to do now to solve this system?
Last edited: