An elementary matrix is one obtained by doing a single row operation to an identity matrix.
The elementary matrix results from doing the row operation to .
The elementary matrix results from doing the row operation to .
The elementary matrix results from doing the row operation to .
Doing a row operation to a matrix has the same effect as multiplying that matrix on the left by the elementary matrix corresponding to .
Let be a row operation and an matrix. Then .
The theorem tells you that doing the row operation to a matrix with three rows is the same as left-multiplying by the elementary matrix . For example, if we do this row operation to we get which is the same as
Elementary matrices are invertible.
Let be a row operation, be the inverse row operation to , and let be an identity matrix. By Theorem 3.8.1, . Because is inverse to , this equals . Similarly, . It follows that is invertible with inverse . ∎
Theorem 3.8.1 combined with Corollary 3.8.2 shows that if results from doing a row operation to , then for some invertible matrix . What about if results from doing a sequence of row operations?
Suppose that is a matrix obtained by doing a sequence of row operations to another matrix . Then for some invertible matrix .