Mathematics The Transformation and Algebra Matrix Question

The transformation Ta of the c-y plane maps r = ai + bj to r’ = Tar = a’i + b’j where)
= A
V
where A is the matrix
21
1 2
Find the cigenvalues of A and show that its eigenvectors are of the form cu and d’y where
c and d are non-zero constants and
u =
(1)
and v=
=(1)
Draw a diagram of the x-y plane and indicate the arrows s =i+j and t = -i+j which
connect the origin to the points with x-y coordinates (1,1) and (-1, 1) respectively, which
correspond to u and v. Also indicate the square which has one corner at the origin and
two of its other corners at the tips of the arrows s and t and indicate the region to which
this square maps under the transformation TA. What is the shape of the transformed
region and what is the ratio of its area to the area of the initial square?