First, since web browsers don't respect tabs, lets clean that up using LaTex:
A has eigenvalues 4 with corresponding eigenvector and 2 with corresponding eigenvector . That means that, if we used those two eigenvectors as basis vectors, we can write the equation as which is equivalent to the two equations x'= 4x and y'= 2y which have solutions and .
Using the eigenvectors as basis had the nice property of separating the equation into two independent equations but now we should also put the intial value in terms of those: which gives the two equations -3a+ b= -6 and a+ b= 1. Subtracting the second equation from the first, -4a= -7 so a= 7/4. Then 7/4+ b= 1 so b= 1- 7/4= -3/4.
That is, must satisfy so C= 7/4, D= -3/4, and .
But, once again, that is in terms of the eigenvectors as basis- those numbers are the coefficients of the eigenvectors, not the "standard basis". To put them in terms of the standard basis,