# Elementary Row Operations & Elementary Matrix

• Elementary Row Operations
• Elementary Matrix
• Relationships

## Elementary Row Operations

There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations):

1. Row Switching
2. Row Multiplication

## Elementary Matrix

An square matrix is called an elementary matrix if it can be obtained from an identity matrix of the same size by performing a single elementary row operation.

## Relationships

If E is an elementary matrix, to apply the elementary row operation to a matrix A, one multiplies A by the elementary matrix on the left (right for column operations). The elementary matrix for any row operation is obtained by executing the operation on the identity matrix.

# Inverse

• Definition
• How to tell if a matrix is invertible
• How to find the inverse

## Definition

For any square matrix A, if there exist a matrix B of the same size such that
$$AB = BA = I$$
then A is said to be invertible and B is called an inverse of A.

If no such B can be found, then A is said to be singular.

Note: all elementary matrices are invertible

## How to tell if a matrix is invertible

A matrix A is invertible if and only if A can be expressed as the product of elementary matrices.

Or, refering to Elementary Linear Algebra with Applications by Howard Anton
, if A is an n by n matrix, then the following statements are equivalent, that is, all true or all false:

1. A is invertible
2. Ax=0 has only the trivial solution
3. The reduced row echelon form of A is an identity matrix
4. A is expressible as a product of elementary matrices

In practice, we decides the invertibility of a square matrix by computing its determinant, i.e.,
A square matrix A is invertible if and only if $det(A) != 0$

## How to find the inverse

Over the year, many methods have been develop to find the inverse of a square matrix. Here I only list a few. For more information, see the wikipage

• Gaussian elimination
• Eigendecomposition
• Analytic solution

### Gaussian elimination

Finds the inverse of a matrix by decomposing a matix into products of elementary matrices. See here for details.

The computational complexity of the method is $O(n^3)$. See here for details.