In each case, Mathematica automatically chooses the best algorithm to use.
Mathematica has state-of-the-art algorithms, here for factoring integers.
This generates a two-dimensional table corresponding to a matrix.
This displays the table in matrix form.
This computes the null space of the matrix.
Here Mathematica solves a nonlinear differential equation numerically.
The answer is an interpolating function which implicitly represents the whole solution. Here is a parametric plot of the solution.