# Matrix intersection of planes

1. Homework Statement

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. Homework 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

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#### AKG

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

but still need help

#### AKG

Homework Helper
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".

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".
another typo on my part i have that c there

#### Hurkyl

Staff Emeritus
Gold Member
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.
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.

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

Help

#### AKG

Homework Helper
"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).

"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).

ok thanks

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