ELEMENTARY ROW OPERATIONS

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In mathematics, 'elementary row operations' are elementary linear transformations on a matrix which preserve matrix equivalence. Thus, elementary row operations do not change the solution set of the system of linear equations represented by a matrix. Elementary operations can be used in Gaussian elimination to reduce a matrix to row echelon form or reduced row echelon form.
A common acronym used in identifying this basic operation dealing with matrices is "ero" or elementary row operations.

Contents

Operations

Row switching

Row multiplication

Row addition

Elementary matrices

Row-switching transformations

Properties

Row-multiplying transformations

Properties

Row-addition transformations

Properties

See also

Operations

There are three types of elementary row operations:

Row switching

A row within the matrix can be switched with another row.
: $R\_i\; leftrightarrow\; R\_j$

Row multiplication

Each element in a row can be multiplied by a non-zero constant.
: $kR\_i$
ightarrow R_i, mbox{where } k
eq 0

Row addition

A row can be replaced by the sum of that row and a multiple of another row.
: $R\_i\; +\; kR\_j$
ightarrow R_i

Elementary matrices

The elementary row operations, like any linear transformation, can be represented in matrix form. These are called 'elementary matrices'.

Row-switching transformations

This transformation, ''T_ij'', switches all matrix elements on row ''i'' with their counterparts on row ''j''. The matrix resulting in this transformation is obtained by swapping row ''i'' and row ''j'' of the identity matrix.
:$$
T_{i,j} = egin{bmatrix} 1 & & & & & & & \ & ddots & & & & & & \ & & 0 & & 1 & & \ & & & ddots & & & & \ & & 1 & & 0 & & \ & & & & & & ddots & \ & & & & & & & 1end{bmatrix},quad
:That is, ''T_ij'' is the matrix produced by exchanging row ''i'' and row ''j'' of the identity matrix.

Properties

:
★ The inverse of this matrix is itself: ''T_ij⁻¹=T_ij''.
:
★ Since the determinant of the identity matrix is unity, det[''T''_''ij''] = −1. It follows that for any conformable square matrix ''A'': det[''T''_''ij''''A''] = −det[''A''].

Row-multiplying transformations

This transformation, ''T_i''(''m''), multiplies all elements on row ''i'' by ''m'' where ''m'' is non zero. The matrix resulting in this transformation is obtained by multiplying all elements of row ''i'' of the identity matrix by ''m''.
:$$
T_i(m) = egin{bmatrix} 1 & & & & & & & \ & ddots & & & & & & \ & & 1 & & & & & \ & & & m & & & & \ & & & & & 1 & & \ & & & & & & ddots & \ & & & & & & & 1end{bmatrix},quad

Properties

:
★ The inverse of this matrix is: ''T_i''(''m'')⁻¹ = ''T_i''(1/''m'').
:
★ The matrix and its inverse are diagonal matrices.
:
★ det[''T''_''i''(m)] = ''m''. Therefore for a conformable square matrix ''A'': det[''T''_''i''(''m'')''A''] = ''m'' det[''A''].

Row-addition transformations

This transformation, ''T_ij''(''m''), subtracts row ''j'' multiplied by ''m'' from row ''i''. The matrix resulting in this transformation is obtained by taking row ''j'' of the identity matrix, and subtracting from it ''m'' times row ''i''.
:$$
T_{i,j}(m) = egin{bmatrix} 1 & & & & & & & \ & ddots & & & & & & \ & & 1 & & & & & \ & & & ddots & & & & \ & & -m & & 1 & & \ & & & & & & ddots & \ & & & & & & & 1end{bmatrix}

Properties

:
★ ''T_ij''(''m'')⁻¹ = ''T_ij''(−''m'') (inverse matrix).
:
★ The matrix and its inverse are triangular matrices.
:
★ det[''T_ij''(''m'')] = 1. Therefore, for a conformable square matrix ''A'': det[''T''_''ij''(''m'')''A''] = det[''A''].

See also

★ Gaussian elimination

★ Linear algebra

★ System of linear equations

★ Matrix (mathematics)