Reduction between transformations
Task number: 3244
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 \(t\ne 0\)\( \begin{pmatrix} a_{11} & … & a_{1n} \\ a_{i1} & … & a_{in} \\ a_{j1} & … & a_{jn} \\ a_{m1} & … & a_{mn} \\ \end{pmatrix}\sim \begin{pmatrix} a_{11} & … & a_{1n} \\ a_{i1} & … & a_{in} \\ ta_{j1} & … & ta_{jn} \\ a_{m1} & … & a_{mn} \\ \end{pmatrix}\sim \begin{pmatrix} a_{11} & … & a_{1n} \\ a_{i1}+ta_{j1} & … & a_{in}+ta_{jn} \\ ta_{j1} & … & ta_{jn} \\ a_{m1} & … & a_{mn} \\ \end{pmatrix}\sim \)
\(\sim \begin{pmatrix} a_{11} & … & a_{1n} \\ a_{i1}+ta_{j1} & … & a_{in}+ta_{jn} \\ a_{j1} & … & a_{jn} \\ a_{m1} & … & a_{mn} \\ \end{pmatrix} \)