## Reduction between transformations

Show that the elementary transformations:
– exchange of two rows, and
– adding of $$t$$-multiple of the $$j$$-th row to the $$i$$-th
could be reduced to transformations:
– multiplication of the $$i$$-th row by a nonzero number $$t$$
– adding the $$j$$-th row to the $$i$$-th.

• #### Resolution

Exchange of the $$i$$-th and $$j$$-th rows:

$$\begin{pmatrix} a_{11} & … & a_{1n} \\ a_{i1} & … & a_{in} \\ a_{j1} & … & a_{jn} \\ a_{m1} & … & a_{mn} \\ \end{pmatrix}\sim \begin{pmatrix} a_{11} & … & a_{1n} \\ a_{i1}+a_{j1} & … & a_{in}+a_{jn} \\ a_{j1} & … & a_{jn} \\ a_{m1} & … & a_{mn} \\ \end{pmatrix}\sim \begin{pmatrix} a_{11} & … & a_{1n} \\ a_{i1}+a_{j1} & … & a_{in}+a_{jn} \\ -a_{i1} & … & -a_{in} \\ a_{m1} & … & a_{mn} \\ \end{pmatrix}\sim$$

$$\sim \begin{pmatrix} a_{11} & … & a_{1n} \\ a_{j1} & … & a_{jn} \\ -a_{i1} & … & -a_{in} \\ a_{m1} & … & a_{mn} \\ \end{pmatrix}\sim \begin{pmatrix} a_{11} & … & a_{1n} \\ a_{j1} & … & a_{jn} \\ a_{i1} & … & a_{in} \\ a_{m1} & … & a_{mn} \\ \end{pmatrix}$$

Adding of the $$t$$-multiple of the $$j$$-th row to the $$i$$-th:
If $$t=0$$, the matrix remains unchanged. We may hence assume that $$t\ne 0$$

$$\begin{pmatrix} a_{11} & … & a_{1n} \\ a_{i1} & … & a_{in} \\ a_{j1} & … & a_{jn} \\ a_{m1} & … & a_{mn} \\ \end{pmatrix}\sim \begin{pmatrix} a_{11} & … & a_{1n} \\ a_{i1} & … & a_{in} \\ ta_{j1} & … & ta_{jn} \\ a_{m1} & … & a_{mn} \\ \end{pmatrix}\sim \begin{pmatrix} a_{11} & … & a_{1n} \\ a_{i1}+ta_{j1} & … & a_{in}+ta_{jn} \\ ta_{j1} & … & ta_{jn} \\ a_{m1} & … & a_{mn} \\ \end{pmatrix}\sim$$

$$\sim \begin{pmatrix} a_{11} & … & a_{1n} \\ a_{i1}+ta_{j1} & … & a_{in}+ta_{jn} \\ a_{j1} & … & a_{jn} \\ a_{m1} & … & a_{mn} \\ \end{pmatrix}$$