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 is obtained by applying elementary transformations to a matrix , then is equivalent to . Denoted by .

Theorem

The elementary row operations that reduce a given matrix to the identity matrix, also transform the identity matrix to the inverse of .

Augmented Matrix

Two matrices are written as a single matrix with a vertical line in-between. Denoted by . Example:

Finding Inverse

Let be a square matrix with order .

• Start with
• Repeatedly perform row transformations (not column) to both matrices until the becomes an identity matrix.
  • Transform all elements outside the main diagonal to .
  • Transform elements on the main diagonal to by multiplying by a constant.
• is .

For singular matrices

They don’t have an inverse, so the process of using elementary row transformations to find will fail in such cases.

Typically occurs because at least one row becomes all zeros during the reduction process, indicating that the matrix has no full rank ().