matrix intersection of planes

canadian_beef

1. The problem statement, all variables and given/known data


Find a necessary condition for the three planes given below to have a line of intersection.

-x +ay+bz=0
ax-y+cz=0
bx+cy-z=0


2. Relevant equations

in order to get a line of intersection between the planes..i know i need one line of the matrix to be [0 0 0|0]


3. The attempt at a solution

well heres the attempt..and its wrong

[ -1 a b | 0
a -1 c | 0
b c -1| 0 ]

=>

[-1 a b | 0
0 (a^2-1) ba+c | 0 (aRow1 + Row2)
0 (ab+c) b^2+1 | 0 ] (brow1 + Row 2)


=>

[ -1 a b | 0
0 a^2 -1 ba+c |0
0 0 2abc +c^2 - a^2 + b^2 +1) |0 ] (ab+c row2- a^2-1 Row1)


then what i did ..by inspection i made 2abc+c^2 -a^2 +b^2 +1 = 0 by letting a=b=1, and c=-1......

but that doesnt work becasue that owuld make plane 1 and 2 the same plane.

i need help

thanks

AKG

The third row in your original matrix should be "b -c -1 | 0" not "b c -1 | 0".

canadian_beef

my bad..edited...i mistyped the question

but still need help

AKG

In your last matrix, the 3rd element of the third row is "2abc +c² - a² + b² +1" but then you start looking at the equation "2ab + c² - a² + b² +1 = 0".

canadian_beef

another typo on my part i have that c there

Hurkyl

So? You weren't asked to find a sufficient condition, you were asked to find a necessary condition.

Incidentally, you have either the polynomial wrong, or the matrix wrong: I think determinants are a simpler approach to the problem than Gaussian elimination.

canadian_beef

Im not sure how to do it the dertiminant way. I do not think my math is wrong so far.

Help

AKG

"a=b=1, c=-1" is a sufficient condition, not a necessary condition. In fact, "2abc + c² - a² + b² +1 = 0" is also just a sufficient condition, not a necessary condition, since it isn't necessary for the third line to be all zeroes (the second line could be all zeroes).

canadian_beef

ok thanks

what would be an example as a necessary conditon and how would i go about finding it

HallsofIvy

If the matrix of coefficients were invertible then the only simultaneous solution to the three equations would be (0, 0, 0), the POINT of intersection of the three planes. In order that the three planes intersect in a line it is necessary that the matrix not be invertible: in other words that the determinant be 0. Find the determinant and set it equal to 0.