Matrix Transformation - A plane mapping onto itself.

[binbagsss]

Find which planes map onto themselves under the matrx M.

M=
1 2 0
0 1 -1
0 2 1
(in enclosed brackets - apologies for the format.).

Attempt:

Consider a plane ax+by+cz=d [1].

M^-1 :
3/3 -2/3 -2/3
0 1/3 1/3
0 -2/3 1/3

(in enclosed bracket).

- use of the inverse so that x,y,z can then be directly subbed into [1].

x= X-2/3(Y)-2/3(Z)
y=(Y)/3+(Z)/3
z=-2(Y)/3 + Z/3

- subbing this into [1] and multiplying throughout by 3 ,gives any plane maps to:

3a(X) + (Y)(-2a+b-2c) + (Z)(-2a+b+c) [2]

Here I am unsure how to interpret the comparison of the co-efficients between [1] and [2] of x,y and z:

3a=a
b= -2a+b-2c => a =c
c= -2a +b+c => 2a=b

I am unsure of what to do next...

 

[lippyka]

any help would be greatly appreciated. The answer is: The plane ax+by+cz=d maps onto itself if a=c and 2a=b.