An augmented matrix is a matrix obtained by adjoining a row or column vector, or sometimes another matrix with the same vertical dimension.
The most common use of an augmented matrix is in the application of Gaussian elimination to solve a matrix equation of the form
(1)

by forming the columnappended augmented matrix
(2)

Augmented matrices can also be used to find a matrix inverse of by forming
(3)

with the identity matrix of appropriate dimension, then using elementary row operations to reduce the result to the form
(4)

The payoff matrix in game theory is a case of a more complicated augmented matrix .