Elementary Matrix
from [wikipedia](https://en.wikipedia.org/wiki/Elementary_matrix)
# Operations
There are three types of elementary matrices, which correspond to three types of row operations (respectively, column 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.
: $$ kR_i \rightarrow R_i,\ \mbox{where } k \neq 0 $$
;Row addition: A row can be replaced by the sum of that row and a multiple of another row.
: $$ R_i + kR_j \rightarrow R_i, \mbox{where } i \neq j $$
If ''E'' is an elementary matrix, as described below, to apply the elementary row operation to a matrix ''A'', one multiplies the elementary matrix on the left, ''E⋅A''. The elementary matrix for any row operation is obtained by executing the operation on the [[identity matrix]].
## Row-switching transformations
The first type of row operation on a matrix ''A'' switches all matrix elements on row ''i'' with their counterparts on row ''j''. The corresponding elementary matrix is obtained by swapping row ''i'' and row ''j'' of the [[identity matrix]].
$$
T_{i,j} =
\begin{bmatrix} 1 & & & & & & & \\ & \ddots & & & & & & \\ & & 0 & & 1 & & \\ & & & \ddots & & & & \\ & & 1 & & 0 & & \\ & & & & & & \ddots & \\ & & & & & & & 1
\end{bmatrix}
\quad
$$
:So ''Tij⋅A'' is the matrix produced by exchanging row ''i'' and row ''j'' of ''A''.
#### Properties
:*The inverse of this matrix is itself: ''Tij−1=Tij''.
:*Since the [[determinant]] of the identity matrix is unity, det[''T''''ij''] = −1. It follows that for any square matrix ''A'' (of the correct size), we have det[''T''''ij''''A''] = −det[''A''].
## Row-multiplying transformations
The next type of row operation on a matrix ''A'' multiplies all elements on row ''i'' by ''m'' where ''m'' is a non-zero [[scalar (mathematics)|scalar]] (usually a real number). The corresponding elementary matrix is a diagonal matrix, with diagonal entries 1 everywhere except in the ''i''th position, where it is ''m''.
$$
D_i(m) = \begin{bmatrix} 1 & & & & & & \\ & \ddots & & & & & \\ & & 1 & & & & \\ & & & m & & & \\ & & & & 1 & & \\ & & & & & \ddots & \\ & & & & & & 1\end{bmatrix}\quad $$
:So D''i(m)⋅A'' is the matrix produced from ''A'' by multiplying row ''i'' by ''m''.
#### Properties
:*The inverse of this matrix is: D''i''(''m'')−1 = D''i''(1/''m'').
:*The matrix and its inverse are [[diagonal matrix|diagonal matrices]].
:*det[D''i''(m)] = ''m''. Therefore for a square matrix ''A'' (of the correct size), we have det[D''i''(''m'')''A''] = ''m'' det[''A''].
## Row-addition transformations
The final type of row operation on a matrix ''A'' adds row ''j'' multiplied by a scalar ''m'' to row ''i''. The corresponding elementary matrix is the identity matrix but with an ''m'' in the (''i,j'') position.
$$
L_{i,j}(m) = \begin{bmatrix} 1 & & & & & & & \\ & \ddots & & & & & & \\ & & 1 & & & & & \\ & & & \ddots & & & & \\ & & m & & 1 & & \\ & & & & & & \ddots & \\ & & & & & & & 1\end{bmatrix}
$$
:So ''Li,j(m)⋅A'' is the matrix produced from ''A'' by adding ''m'' times row ''j'' to row ''i''.
### Properties
:*These transformations are a kind of [[shear mapping]], also known as a ''transvections''.
:*''Lij''(''m'')−1 = ''Lij''(−''m'') (inverse matrix).
:*The matrix and its inverse are [[triangular matrix|triangular matrices]].
:*det[''Lij''(''m'')] = 1. Therefore, for a square matrix ''A'' (of the correct size) we have det[''L''''ij''(''m'')''A''] = det[''A''].
:*Row-addition transforms satisfy the [[Steinberg relations]].