It is not necessary to find the matrix representing T. "Tfixes the line y= 4xpointwise" means that if T does not change points on that line; if y= 4x, then T(x, y)= (x, y) or T(x, 4x)= (x, 4x)= 1(x, 4x). ThatTellsyou that 1 is an eigenvalue with eigenvector (1, 4). Saying that T " maps every point on the line x + 4y = 0 to the orign" means that T(-4y, y)= (0, 0)= 0(-4y, y) and that tells you that 0 is an eigenvalue with eigen vector (-4, 1).

But, in fact, it is not hard to find the the matrix representing T. The two lines, y= 4x, and x= -4y are perpendicular (their slopes are 4 and -1/4) so we could use (1, 4) and (-4, 1) as basis vectors- every vector, (x, y), can be written as a linear combination of (1, 4) and (-4, 1). (x, y)= a(1, 4)+ b(-4, 1)= (a- 4b, 4a+ b) so a- 4b= x and 4a+ b= y. Solve for a and b: 17a= x+ 4y so a= (x+ 4y)/17 and 17b= y- 16x so b= (y- 16x)/17. In particular, (1, 0)= 1/17(1, 4)- 16/17(-4, 1) and (0, 1)= 4/17(1, 4)+ 1/17(-4, 1).

T(1, 0)= 1/17 T(1, 4)- 16/17(-4, 1)= 1/17 (1, 4)= 1/17(1, 0)+ 4/17(0, 1) and T(0, 1)= 4/17 T(1, 4)+ 1/16 T(-4, 1)= 4/17 T(1, 4)= 4/17(1, 0)+ 16/17(0, 1). The coefficients in those equations give the columns of the matrix representation of T:

But, as I said, you don't need to find that matrix representation- the information given tells you directly what the eigenvalues and eigenvectors are.