Question

If $A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right],B=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]$, find AB+BA-2B

Answer 1

$A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right],B=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]$

$AB=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 3\\ 1& 3\end{array}\right]$

$BA=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]=\left[\begin{array}{cc}3& 3\\ 1& 1\end{array}\right]$

$2B=2\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}2& 4\\ 0& 2\end{array}\right]$

$AB+BA-2B=\left[\begin{array}{cc}1& 3\\ 1& 3\end{array}\right]+\left[\begin{array}{cc}3& 3\\ 1& 1\end{array}\right]-\left[\begin{array}{cc}2& 4\\ 0& 2\end{array}\right]$

$=\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right]$