From a pedagogical point of view, examples of matrix inversion are invariably coupled with the use of the

determinant and 3x3 matrices.

From a

numerical analysis point of view, it is almost never desirable to actually compute the inverse of a matrix. On a practical level, the inverse of a matrix is almost never what you want to compute for "real" matrices. The reasons are as follows:

- The determinant will overflow IEEE754 doubles for "real" sized matrices
- The inversion of a matrix takes O(n
^{3}) operations. One typically wants to solve for a vector x=A^{-1}b or matrix C=A^{-1}B. In this case, multiplying by the inverse is far more computationally expensive than performing an LU factorisation, followed by right hand solves.

There are instances where a generalized matrix inversion is unavoidable. In this case, one should still avoid the use of the

determinant. As suggested by

koala's post, one can perform a factorisation (

LU or

Cholesky, for example), and employ the columns of the

identity matrix as multiple right hand sides.