Elementary Transformations
- Interchange of any columns or rows
- Addition of multiple of any row or column to any other row or column
- Multiplication of each element of a column or a row by a non-zero constant
When a matrix
Theorem
The elementary row operations that reduce a given matrix
Augmented Matrix
Two matrices are written as a single matrix with a vertical line in-between.
Denoted by
Inverse using elementary row transformations
Let
- Start with
- Repeatedly add row transformations (not column) to both of the matrices
until the
becomes an identity matrix. - Convert all elements outside the main diagonal to
. - Convert elements on the main diagonal to
by multiplying by a constant.
- Convert all elements outside the main diagonal to
- When
is an identity matrix, is .