I was able to find a definition of the difference between an analytic solution and a numerical method. Solving something analytically means thinking through the problem and solving it by hand. Solving something numerically usually means iterating a formula to get an approximate solution. Solving numerically is done when the problem in question is too complicated to solve analytically.
I spent most of the day reading up on matrix determinants. A few interesting things to note are: Only square matrices have determinants. The determinant of a matrix is the area of the parallelogram(2D)/parallelepiped(3D) defined by the row vectors of the matrix. The sign of the determinant can be used to perform basic collision tests, such as checking which side of a line a point is on. The sign of the determinant also tells you whether or not the matrix can be inverted (if the determinant is 0, the matrix does not have an inverse).