Final answer:
If a^4 equals 3i, then a is indeed invertible because a non-zero complex number always has an inverse. This implies that there exists a multiplicative inverse for a which, when raised to the fourth power, yields 1.
Step-by-step explanation:
If a^4 = 3i, then a is invertible? The answer is True. A number is invertible if there exists another number such that when multiplied by the original number, it results in the multiplicative identity, which is 1 for real numbers and in general, for the set of complex numbers. Since a^4 = 3i is a non-zero complex number, there must exist some other complex number that when raised to the fourth power equals the multiplicative inverse of 3i. Hence, a is invertible because we can find an a^-1 such that aa^-1 = 1. This also implies that a has a non-zero determinant if we consider a as a matrix representation.