5
$\begingroup$

On solving $$ 2x - 4y + z = 0 $$ $$ x + y - 4z = 0 $$ $$ x - y - z = 0 $$ I get $$ y = 0.6 x $$ $$ z = 0.4 x$$ I thought that there was a rule of thumb that you need as many independent equations as the number of unknowns. So, why don't I get a unique solution in this case?

$\endgroup$
  • 1
    $\begingroup$ Your homogeneous system has the trivial solution $x = y = z = 0$. If it was unique, that would be quite boring, wouldn't it? But as @user2345215 explained, it is not unique because your equations are not linearly independent. $\endgroup$ – GDumphart Nov 22 '14 at 15:36
3
$\begingroup$

Hint: The matrix of the coefficients of the given system of equations is $$A=\begin{pmatrix}2&-4&1\\1&1&-4\\1&-1&-1\end{pmatrix}$$ with $det(A)=0$ so the equations are not independent. Therefore the fact that you do not get a unique solution is not a contradiction to what you (correctly) know.

$\endgroup$
5
$\begingroup$

Because $$\begin{pmatrix}2&-4&1\\1&1&-4\\1&-1&-1\end{pmatrix}\sim\begin{pmatrix}0&-2&3\\0&2&-3\\1&-1&-1\end{pmatrix}\sim\begin{pmatrix}0&0&0\\0&2&-3\\1&-1&-1\end{pmatrix},$$ so the system isn't independent.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.