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 [23]. An initial introduction to matrices
and linear algebra can be found in [46].