Question

State true or false:

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

• A. True
• B. False

Answer 1

The correct option is B False

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

$(A+B{)}^2=A^2+AB+BA+B^2=A^2+2AB+B^2$ is true only when $AB=BA$

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

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

$AB\ne BA$

$\therefore$ Given statement is false.