- #1

binbagsss

- 1,261

- 11

**Find which planes map onto themselves under the matrx M.**

__M=__**1 2 0**

0 1 -1

0 2 1

0 1 -1

0 2 1

(in enclosed brackets - apologies for the format.).

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

3/3 -2/3 -2/3

0 1/3 1/3

0 -2/3 1/3

__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...