Consider the linear system of equations
in matrix form:
This can be written in higher level form as
where denotes the two-by-two matrix above, and and denote the two-by-one vectors. The solution to this equation is then
The general two-by-two matrix inverse is given by
and the inverse exists whenever (which is called the determinant of the matrix ) is nonzero. For larger matrices, numerical algorithms are used to invert matrices, such as used by Matlab based on LINPACK . An initial introduction to matrices and linear algebra can be found in .