How can I solve this system of linear equations?

Using either Gaussian or Gauss-Jordan elimination, find the value(s) of k, if any, for which the following system will have

(i) no solution

(ii) a unique solution, and

(iii) infinitely many solutions

x − 3z = −3

−2x − λy + z = 2

x + 2y + λz = 1

Thanks for any help

Re: How can I solve this system of linear equations?

There is no "k" in your equations. I assume you're looking for values of lambda, not "k", right?

Can you set this up as a matrix? Can you row-reduce that matrix?

Re: How can I solve this system of linear equations?

if $\displaystyle \lambda $ = -5 (no solution) or 2 (infinitely many soln), for every other value, a unique solution exists. . I cheated and used determinant.

Re: How can I solve this system of linear equations?

Sorry, do you mind briefly explaining how you got these answers, just I get what I'm asked to do, just confused by the process..

Re: How can I solve this system of linear equations?

I took the determinant of

$\displaystyle A = \begin{bmatrix} 1 & 0 & -3 \\ -2 & -\lambda & 0 \\ 1 & 2 & \lambda \end{bmatrix} $

to get Det(A) = $\displaystyle - \lambda^{2} - 3 \lambda + 10 $ and i used the quadratic formula to get the roots, $\displaystyle \lambda = -5, 2 $. A system has zero or infinitely many solution only when Det(A) = 0. so Det(A) = 0, when $\displaystyle \lambda = -5, 2 $. I then found that for $\displaystyle \lambda = -5 $ using row reducing, that the system didint have a solution, and again for $\displaystyle \lambda = 2 $, i found out by row reducing that it had infinite soln.

Re: How can I solve this system of linear equations?

Makes so much sense, thank you !

Re: How can I solve this system of linear equations?

Sorry Jakncoke,

I was just confirming, would the values for λ be 5, -2 ?

because of the negative in front of -λ^2 - 3λ + 10

-(λ - 5)(λ + 2)

Re: How can I solve this system of linear equations?

Quote:

Originally Posted by

**cellae** Sorry Jakncoke,

I was just confirming, would the values for λ be 5, -2 ?

because of the negative in front of -λ^2 - 3λ + 10

-(λ - 5)(λ + 2)

$\displaystyle (\lambda - 5)(\lambda + 2) = \lambda^{2} + 2 \lambda - 5 \lambda - 10 = \lambda^{2} + -3 \lambda - 10 $ negative of that is $\displaystyle -\lambda^{2} + 3 \lambda + 10 $ which is not the determinant (the 3 has to be -3) so root ($\displaystyle \lambda $ is as stated -5, and 2.

Re: How can I solve this system of linear equations?