Solve for Alpha(a): 3 Equations with Unknowns in Matrix - Need Help!

[Martinuk]

Homework Statement

the following 3 equations are given, show that alpha(a) has two non trival solutions, then solve for the whole thing:

Homework Equations

ax - 3y + ( 1+a) z = 0
2x + y - az = 0
(a+2)x - 2y + az = 0


i think i may be missing something but i can't seem to even get started on this problem,
thanks

 

[HallsofIvy]

Homework Statement

the following 3 equations are given, show that alpha(a) has two non trival solutions, then solve for the whole thing:

Homework Equations

ax - 3y + ( 1+a) z = 0
2x + y - az = 0
(a+2)x - 2y + az = 0


i think i may be missing something but i can't seem to even get started on this problem,
thanks

Well, what HAVE you done? You mention "unknowns in a matrix" so presumably you know how to set this up as a matrix equation. With A as the matrix of coefficients, X as the column matrix <x, y, z> and 0 as the column matrix <0, 0, 0>, your equation is AX= 0. The unique solution will be X= A^{-1}0= 0 as long as A has an inverse.

So the question is, "for what values of a does this matrix of coefficients not have an inverse?" I don't know what methods of finding an inverse matrix you have learned but you might try row-reducing the matrix and see what values of a prevent you doing that. Or determine what values of a make the determinant of the matrix 0.