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 single elementary row operation.
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.