Matrix Inverse

A Matrix ${\hbox{\sf A}}$ has an inverse Iff the Determinant $\vert{\hbox{\sf A}}\vert\not = 0$. For a $2\times 2$ Matrix

$${\hbox{\sf A}}\equiv \left[{\matrix{a & b\cr c & d\cr}}\right],$$ (1)

the inverse is

$${\hbox{\sf A}}^{-1} = {1\over \vert{\hbox{\sf A}}\vert} \lef... ... = {1\over ad-bc} \left[{\matrix{d & -b\cr -c & a\cr}}\right].$$ (2)

For a $3\times 3$ Matrix,

$${\hbox{\sf A}}^{-1} = {1\over \vert{\hbox{\sf A}}\vert} \le... ...a_{11} & a_{12}\cr a_{21} & a_{22}\cr}\right\vert\cr}}\right].$$ (3)

A general $n\times n$ matrix can be inverted using methods such as the Gauss-Jordan Elimination, Gaussian Elimination, or LU Decomposition.

The inverse of a Product ${\hbox{\sf A}}{\hbox{\sf B}}$ of Matrices ${\hbox{\sf A}}$ and ${\hbox{\sf B}}$ can be expressed in terms of ${\hbox{\sf A}}^{-1}$ and ${\hbox{\sf B}}^{-1}$. Let

$${\hbox{\sf C}}\equiv {\hbox{\sf A}}{\hbox{\sf B}}.$$ (4)

Then

$${\hbox{\sf B}}={\hbox{\sf A}}^{-1}{\hbox{\sf A}}{\hbox{\sf B}}={\hbox{\sf A}}^{-1}{\hbox{\sf C}}$$ (5)

and

$${\hbox{\sf A}}={\hbox{\sf A}}{\hbox{\sf B}}{\hbox{\sf B}}^{-1}={\hbox{\sf C}}{\hbox{\sf B}}^{-1}.$$ (6)

Therefore,

$${\hbox{\sf C}}={\hbox{\sf A}}{\hbox{\sf B}}=({\hbox{\sf C}}{... ...x{\sf C}}{\hbox{\sf B}}^{-1}{\hbox{\sf A}}^{-1}{\hbox{\sf C}},$$ (7)

so

$${\hbox{\sf C}}{\hbox{\sf B}}^{-1}{\hbox{\sf A}}^{-1}={\hbox{\sf I}},$$ (8)

where I is the Identity Matrix, and

$${\hbox{\sf B}}^{-1}{\hbox{\sf A}}^{-1}={\hbox{\sf C}}^{-1}=({\hbox{\sf A}}{\hbox{\sf B}})^{-1}.$$ (9)

See also Matrix, Matrix Addition, Matrix Multiplication, Moore-Penrose Generalized Matrix Inverse, Strassen Formulas

References

Ben-Israel, A. and Greville, T. N. E. Generalized Inverses: Theory and Applications. New York: Wiley, 1977.

Nash, J. C. Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 24-26, 1990.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Is Matrix Inversion an $N^3$ Process?'' §2.11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 95-98, 1992.