Question

If $A=\left|\begin{array}{c}\begin{array}{c}-2\\ 4\end{array}\\ 5\end{array}\right|,B=\left|\begin{array}{c}\begin{array}{ccc}1& 3& -6\end{array}\end{array}\right|$, State whether ist is true or false ${\left(AB\right)}^1=B^1{A}^1$

• A. True
• B. False

Answer 1

The correct option is B False

$A=\left[\begin{array}{c}-2\\ 4\\ 5\end{array}\right],B=\left[\begin{array}{ccc}1& 3& -6\end{array}\right]$

Write $(AB{)}^1=B^1{A}^1{A}^1=A=\left[\begin{array}{ccc}-2& 4& 5\end{array}\right]B^1=\left[\begin{array}{c}1\\ 3\\ -6\end{array}\right]AB=\left[\begin{array}{c}2\\ 4\\ 5\end{array}\right]\left[\begin{array}{ccc}1& 3& -6\end{array}\right]=\left[\begin{array}{c}-2\times 1\\ 4\times 3\\ 5(-6)\end{array}\right]=\left[\begin{array}{c}-12\\ 12\\ -30\end{array}\right]\to (i\left)\right(AB{)}^1=\left[\begin{array}{ccc}-2& 4& 5\end{array}\right]B^1{A}^1=\left[\begin{array}{c}1\\ 3\\ -6\end{array}\right]\left[\begin{array}{ccc}-2& 4& 5\end{array}\right]=\left[\begin{array}{c}1\times -2\\ 3\times 4\\ -6\times 5\end{array}\right]=\left[\begin{array}{c}-2\\ 12\\ -30\end{array}\right]\to \left(ii\right)\left(i\right)\&\left(ii\right)(AB{)}^1\ne B^1{A}^1$

Hence False