An ** elementary row operation** is a linear transformation , where the transformation matrix performs one of the following on :

If is the identity matrix , the transformed matrix is called an ** elementary matrix**, which is denoted by in place of . In other words, an elementary matrix is a square matrix that is related to an identity matrix by a

*elementary row operation.*

__single__For example,

are elementary matrices, where and are obtained from by

**Type 1**. Swapping rows 1 and 2 of .

**Type 2**. Multiplying row 2 of by 7

**Type 3**. Adding 4 times row 2 of to row 1 of

respectively.

Interestingly, itself is a transformation matrix if because . Therefore, when we multiply by a matrix , we are performing an elementary row operation on . For example,

An elementary matrix of dimension has an inverse if , where the inverse is a matrix that reverses the transformation carried out by . Every elementary matrix has an inverse because

**Type 1**. Two successive row swapping operations of a matrix returns , i.e. . Comparing with , we have .

**Type 2**. It is always possible to satisfy when and differ by one diagonal matrix element , with , and .

**Type 3**. It is always possible to satisfy when and differ by one matrix element , where and with , , and .

Thus, * all elementary matrices have corresponding inverses, which are themselves elementary matrices*. For example, the inverses of and are

Finally, ** a non-singular matrix can always be expressed as a product of elementary matrices**. The proof is as follows:

Let . Since every elementary matrix is non-singular, we can multiply the inverses of the elementary matrices successively on the left of to give:

Similarly, we can multiply the inverses of the elementary matrices successively on the right of to give:

Combining eq3 and eq4, we have , where , which completes the proof.