nontrivial matrix: find all values for which Bx = 0 has....

[luckyc1423]

\(\displaystyle \begin{array}{l}
{\rm Find all values for which the matrix equation Bx = 0 has a nontrivial solution,} \\
{\rm where} \\
{\rm B = }\left[ \begin{array}{l}
{\rm 1 - }\lambda {\rm 1} \\
{\rm - 2 4 - }\lambda \\
\end{array} \right] \\
\end{array}\)

 

[pka]

Here is a big hint: If \(\displaystyle \left| B \right| \not= 0\) then there will be a non-trivial solution.

 

[luckyc1423]

Well I knew that, but something is just hanging me up, we did this in class and he gave us facts, like the one you just posted, but for something reason my brain just is unwilling to cooperate

 

[pka]

It seems to me as if you do not know how to find the determinate .
\(\displaystyle \L
\left| {\begin{array}{cc}
{1 - \lambda } & 1 \\
{ - 2} & {4 - \lambda } \\
\end{array}} \right| = 6 - 5\lambda + \lambda ^2 .\)

 

[luckyc1423]

Ur right, we just learned this in class today and these problems are due tomorrow, this prof. doesnt speak english well so its hard for me to pick him up. Thanks for your help. I am sure I will be able to do the steps to lead up to your answer.

 

[soroban]

Re: nontrivial matrix: find all values for which Bx = 0 has.

Hello, luckyc1423!

pak is absolutely correct.
To have a nontrivial solution, the determinant must be nonzero.
I'll walk you through it . . .

Find all values for which the matrix equation \(\displaystyle Bx\,=\, 0\) has a nontrivial solution,

where: \(\displaystyle \,B\:=\:\begin{bmatrix}1-\lambda & 1 \\ -2 & 4-\lambda\end{bmatrix}\)

The determinant of \(\displaystyle B\) is: \(\displaystyle \,D\;=\;(1-\lambda)(4-\lambda)\,-\,(1)(-2) \;= \;6\,-\,5\lambda\,+\,\lambda^2\)


Now when is \(\displaystyle D\) not equal to 0 ?

An easier question is: when does \(\displaystyle D\) equal 0 ?

We have: \(\displaystyle \,\lambda^2\,-\,5\lambda\,+\,6\:=\:0\)

\(\displaystyle \;\;\)which factors: \(\displaystyle \,(\lambda\,-\,2)(\lambda\,-\,3)\:=\:0\)

\(\displaystyle \;\;\)and has roots: \(\displaystyle \,\lambda\,=\,2,\;3\)


Threfore, \(\displaystyle Bx\,=\,0\,\) has a nontrival solution for all \(\displaystyle \lambda\,\neq\,2\) or \(\displaystyle 3\)

 

[luckyc1423]

Thanks alot for the help, I guess I was confused on the non-trivial part, but with your help it makes everything seem so easy, once again thanks!