why strange solutions obtained?

Maybe it would help if you explain why you think that the result is wrong and what result (Matlab screenshot) you expect.

You want the system to be solved for a time interval up to 10^4 but you allow just (the default) 10^3 steps. A step width of 10 time units maybe is too large for a meaningful result. You may consider to solve for a smaller time interval and/or increase the number of iteration steps (thats an additional argument of odesolve).

I tried and the result is that A(t) goes down to zero right at the beginning when the precison is enhanced that way.

It looks like A is transferred amost immediatly to E, which is, looking at your coefficients k, no big surprise. Could it be a problem with numerical precision or what result did you expect? Your coefficients are rather big and k3 and k5 seem to be nearly neglectable compared to the others. Are you sure you used the same equations and the same coefficients in Matlab?

Can it be that it should be