Question

Sa se verifice AB=BA in matricea :
A= 1 2 B= 0 1
0 1 1 -1
matricea intre paranteze rotunde

Answer 1

$A=\begin{pmatrix} 1&2 \\ 0& 1 \end{pmatrix}$

$B=\begin{pmatrix} 0&1 \\ 1& -1 \end{pmatrix}$

$AB=\begin{pmatrix} 1&2 \\ 0& 1 \end{pmatrix}\times\begin{pmatrix} 0& 1\\ 1 & -1 \end{pmatrix}$

$AB=\begin{pmatrix} 1\times0+2\times1 & 1\times1+2\times(-1)\\ 0\times0+1\times1 & 0\times1+1\times(-1) \end{pmatrix}$

$AB=\begin{pmatrix} 2 & -1\\ 1 & -1 \end{pmatrix}$

$BA=\begin{pmatrix} 0&1 \\ 1& -1 \end{pmatrix}\times\begin{pmatrix} 1& 2\\ 0 & 1 \end{pmatrix}$

$BA=\begin{pmatrix} 0\times1+1\times0 & 0\times2+1\times1\\ 1\times1+(-1)\times0 & 1\times2+(-1)\times1 \end{pmatrix}$

$BA=\begin{pmatrix} 0 & 1\\ 1 & 1 \end{pmatrix}$

$= > \begin{pmatrix} 2&-1 \\ 1& -1 \end{pmatrix} \neq \begin{pmatrix} 0 & 1\\ 1 & 1 \end{pmatrix}$

$= > AB \neq BA$