Given that a matrix inverse keeps a record of the cumulative row operations performed to bring a matrix to RREF, it is also sometimes useful to represent a single row operation as an elementary matrix. An elementary matrix for a single row operation can be found by performing that operation on the 
-by-
identity matrix where 
is the number of equations in the system. For example 
is represented by the following matrix:
Once again, the row operations for the matrix used in previous subsections will be used to demonstrate elementary matrices:
Multiplying all of these matrices together at once gives the matrix inverse. Note the order of multiplication. The order of row operations corresponds to a right-to-left order in multiplication:
-by- identity matrix where is the number of equations in the system. For example is represented by the following matrix:Excercise 5-6.
Write the following elementary matrices:
- The row operation
for a system with 
equations and 
unknowns. - The row operation
for a system with 
equations and 
unknowns.
elementary matrix is found by the following: 
