Pure Maths---Matrix2

since the invariant line is x-2y=0, [0 0] is on the line

let the matrix be [ a b ; c d ] ,
we have
[ x' y'] =[ a b ; c d ] [x y]

since [3,-1] map to [ 0 -5/2], then

0 = 3a-b
-2.5 = 3c-d

and x-2y=0 is invariant line, pick any point of the line, say (2,1), we have
2 = 2a+b
1 = 2c+d

than solve 4 equations with 4 unknown.
the matrix is
[ 2/5 6/5; -3/10 8/5]

*****************

the invert matrix of a) is
[ 8/5 -6/5; 3/10 2/5]
thus,
x= 8/5x'-6/5 y'
y = 3/10 x' + 2/5 y'

sub into the T
(8/5x'-6/5 y')^2+16( 3/10 x' + 2/5 y')^2=4
simplify
x'x'+y'y'=1

which is an circle of radiu 1.