Product of a 2x2 and a general matrices
Task number: 2421
Determine products \(\mathbf A\mathbf B_i\) and \(\mathbf B_i\mathbf A\), for matrices
\(\mathbf A=\begin{pmatrix} a_{1{,}1} & a_{1{,}2} \\ a_{2{,}1} & a_{2{,}2} \end{pmatrix}\), \(\mathbf B_1=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\), \(\mathbf B_2=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\) a \(\mathbf B_3=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\).
Which transformations of \(\mathbf A\) these products provide?
Resolution
\( \mathbf A\mathbf B_1= \begin{pmatrix} a_{1{,}1} & 0 \\ a_{2{,}1} & 0 \end{pmatrix} \) – corresponds of the choice of the first column.
\( \mathbf B_1\mathbf A= \begin{pmatrix} b_{1{,}1} & b_{1{,}2} \\ 0 & 0 \end{pmatrix} \) – corresponds of the choice of the first row.
\( \mathbf A\mathbf B_2= \begin{pmatrix} 0 & a_{1{,}1} \\ 0 & a_{2{,}1} \end{pmatrix}\) – chooses the first column and puts it as the second.
\( \mathbf B_2\mathbf A= \begin{pmatrix} a_{2{,}1} & a_{2{,}2} \\ 0 & 0 \end{pmatrix}\) – chooses thesecond row and puts it as the first.
\( \mathbf A\mathbf B_3= \begin{pmatrix} a_{1{,}2} & a_{1{,}1} \\ a_{2{,}2} & a_{2{,}1} \end{pmatrix}\) a \(\mathbf B_3\mathbf A= \begin{pmatrix} a_{2{,}1} & a_{2{,}2} \\ a_{1{,}1} & a_{1{,}2} \end{pmatrix} \) – the first product swaps columns, the other rows.
Products \(\mathbf B\mathbf A\) correspond to the row transformations of \(\mathbf A\), while products \(\mathbf A\mathbf B\) yield column transformations of \(\mathbf A\).
