## Product of a 2x2 and a general matrices

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 $$\mathbf A$$.