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Sahithyan's S1

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:

Inverse using elementary row transformations

Let be a square matrix with order .

  • 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.
  • When is an identity matrix, is .