Elementary row operation and elementary matrix

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


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.


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