# System of homogeneous equations

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1. Mar 17, 2015

I got three equations:
l-cm-bn=0
-cl+m-an=0
-bl-am+n=0
In my textbook, its written "eliminating l, m, n we get:"
$$\begin{vmatrix} 1& -c& -b\\ -c& 1& -a\\ -b& -a& 1\\ \end{vmatrix}=0$$
but if I take l, m, n as variables and since $l=\frac{\Delta_1}{\Delta}$ (Cramer's rule) and $\Delta_1=0$, then if $\Delta=0$,you get an indeterminate form.
Is the expression given in my textbook correct?

2. Mar 17, 2015

### SteamKing

Staff Emeritus
It's not clear what is going on in your text book. To me, it looks like l, m, and n are the unknown variables for this system. If elimination were correctly carried out on the matrix of coefficients, then you would be left with only 1's on the main diagonal and only zeros to the lower left of the main diagonal.

In any event, the solution of a system homogeneous linear equations requires special consideration. If the determinant of the matrix of coefficients is not equal to zero, then l = m = n = 0 is the only solution to the system. If the determinant of the matrix of coefficients equals zero, there is an infinite number of solutions.

http://en.wikipedia.org/wiki/System_of_linear_equations

3. Mar 17, 2015

"If the planes x=cy+bz, y=az+cx and z=bx+ay pass through a line, then the value of $a^2+b^2+c^2+2abc=$?