# Elementary Matrices

The transformations we perform on a system or on the corresponding augmented matrix, when we attempt to solve the system, can be simulated by matrix multiplication. More precisely, each of the three transformations:

- multiplication of a row by a nonzero constant
*k*; - interchanging two rows;
- addition of a constant
*k*times one row to another;

An **elementary matrix** is a matrix which differs from the identity matrix by one single elementary row operation.

Since there are three elementary row transformations, there are three different kind of elementary matrices.
If we let **A** be the matrix that results from **B** by performing
one of the operations in the above list, then the matrix **B** can be recovered from **A** by performing
the corresponding operation in the following list:

- multiplication of the same row by constant
*1/k*; - interchanging the same two rows;
- if
**A**results by adding*k*times row*i*of**B**to row*j*, then add*-k*times row*j*to row*j*.

The elementary matrices generate the general linear group of invertible matrices. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form. They are also used in Gauss-Jordan elimination to further reduce the matrix to reduced row echelon form.

```
{{1, 0, 0}, {0, 1, 0}, {-2, 0, 1}}.{{1, 2, 3, 4}, {1, -1, -2, -3}, {2, 3, 1, -1}}
```

Theorem: If the elementary matrix E results from performing a certain row operation
on the identity *n*-by-*n* matrix and if **A** is an
\( n \times m \) matrix, then the product **E A** is the
matrix that results when this same row operation is performed on **A**. ■

Theorem: The elementary matrices are nonsingular. Furthermore, their inverse is also an elementary matrix. That is, we have:

- The inverse of the elementary matrix which interchanges two rows is itself.
- The inverse of the elementary matrix which simulate \( \left( k\,R_i \right) \leftrightarrow \left( R_i \right) \) is the elementary matrix which simulates \( \left( k^{-1} R_i \right) \leftrightarrow \left( R_i \right) . \)
- The inverse of the elementary matrix which simulates \( \left( R_j + k\,R_i \right) \leftrightarrow \left( R_j \right) \) is the elementary matrix which simulates \( \left( R_j - k^{-1} R_j \right) \leftrightarrow \left( R_i \right) . \) ■

*n*-dimensional real space, and it is called the

**standard basis**. Its dimension is

*n*.

*i*-th row and

*j*-th column. Then the set \( {\bf M}_{i,j} \ : \ 1 \le i \le m , \ 1 \le j \le n \) is a basis for the set of all such real matrices. Its dimension is

*mn*.

*n*. It has dimension

*n*+1. ■

If the vectors \( \left\{ {\bf u}_1 , {\bf u}_2 , \ldots , {\bf u}_n \right\} \)
form a basis for a vector space *V*, then every vector in *V* can be uniquely expressed in the form
switching rows
1
and
2
{\bf v} = \alpha_1 {\bf u}_1 + \alpha_2 {\bf u}_2 + \cdots + \alpha_n {\bf u}_n
\]