Elementary Operations on a Matrix

Elementary Row Operations:

  1. Row Replacement (or Row Addition):

    • Replace one row with the sum of itself and a multiple of another row.
    • For row i and row j in a matrix A, Ri=Ri+k×Rj.
  2. Row Interchange:

    • Interchange (swap) two rows within a matrix.
    • Exchange rows i and j in matrix A.
  3. Row Scaling (or Row Multiplication):

    • Multiply a row by a non-zero scalar.
    • For row i in matrix A, Ri=k×Ri.

Elementary Column Operations:

  1. Column Replacement (or Column Addition):

    • Replace one column with the sum of itself and a multiple of another column.
    • For column i and column j in a matrix A, Ci=Ci+k×Cj.
  2. Column Interchange:

    • Interchange (swap) two columns within a matrix.
    • Exchange columns i and j in matrix A.
  3. Column Scaling (or Column Multiplication):

    • Multiply a column by a non-zero scalar.
    • For column i in matrix A, Ci=k×Ci.

Example:

Consider a matrix A:

A=[2354]
  1. Row Replacement: R2=R25×R1
[2354][232311]
  1. Row Interchange: Swap rows 1 and 2
[2354][5423]
  1. Row Scaling: R1=12×R1
[2354][13254]

Example:

Consider a matrix A:

A=[2354]
  1. Column Replacement: C2=C23×C1
[2354][27511]
  1. Column Interchange: Swap columns 1 and 2
[2354][3245]
  1. Column Scaling: C1=12×C1
[2354][1354]

Properties and Significance:

  • Elementary operations do not change the solutions of a system of linear equations represented by the matrix.

  • They are fundamental in various matrix operations, including row reduction, finding inverses, and solving systems of equations.