## Elementary matrices

Let be the identity matrix. The matrix  obtained by either

1. adding a multiple of the -th row of  by to the -th row of , or
2. interchanging the -th row and the -th row of , or
3. multiplying the -th row of  by ,
is called an elementary matrix. Given an matrix , the matrix multiplication  results in performing the corresponding basic row operation on . Furthermore, the inverse matrix  of the elementary matrix  is also the elementary matrix obtained respectively by
1. adding a multiple of the -th row of  by to the -th row of ,
2. interchanging the -th row and the -th row of  (thus, the same matrix),
3. multiplying the -th row of  by .

EXAMPLE 7. Find the elementary matrix corresponding to each of the following row operations:

1. Add a multiple of the 1st row by -4 to the 3rd row.
2. Interchange the 1st and 2nd row.
3. multiplying the 3rd row by 5.

EXAMPLE 8. Find the inverse of

Finding . Let be a square matrix. Suppose that . Then we have a series of basic row operations which reduces  to . Or, equivalently we have a series of elementary matrices  so that . Comparing

and

we have .

Department of Mathematics