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Elementary matrices

Let be the identity matrix.
The matrix obtained by either

- adding a multiple of the -th row of by to the -th row
of , or
- interchanging the -th row and the -th row of , or
- 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
- adding a multiple of the -th row of by to the -th row
of ,
- interchanging the -th row and the -th row of
(thus, the same matrix),
- multiplying the -th row of by .

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

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

*Last modified: 2005-10-21*