In this paper we derive a number of invariant tests for the problem of testing linear hypotheses. The power functions of these tests are studied and it turns out to depend on die value of r = rank(XlZ) (where X and Z are the given regressor matrices) whether the tests have level a, are unbiased and possess certain other desirable properties. The required computations in order to use the tests in practice are given. We also derive large sample approximations to the critical values and the p-values of the tests.