Complexity of matrix inverse via Gaussian elimination
No, it's not true.  You can use Gaussian elimination to invert a matrix in $O(n^3)$ time, but there are other algorithms that are even faster.  The complexity of a problem is the running time of the fastest algorithm for that problem.  Therefore, it is not correct that the complexity of matrix inversion is $\Theta(n^3)$.

See https://en.wikipedia.org/wiki/Gaussian_elimination#Computational_efficiency, https://en.wikipedia.org/wiki/Matrix_multiplication_algorithm, https://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations#Matrix_algebra, https://scicomp.stackexchange.com/q/22105/4274, Complexity class of Matrix Inversion.