If-A-2-1-3-4-1-0-and-B-1-1-0-2-5-0-verify-that-AB-B-A-

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Question Number 7 by user1 last updated on 25/Jan/15

$$\mathrm{If}A=\left[\begin{array}{ccc}2& 1& 3\\ 4& 1& 0\end{array}\right]andB=\left[\begin{array}{cc}1& -1\\ 0& 2\\ 5& 0\end{array}\right],$$$$\mathrm{verify}\mathrm{that}{\left(\mathrm{A}B\right)}^{\prime }=B^{\prime }{A}^{\prime }$$

Answered by user1 last updated on 30/Oct/14

$$\mathrm{We}\mathrm{have}$$$$AB=\left[\begin{array}{ccc}2& 1& 3\\ 4& 1& 0\end{array}\right]\cdot \left[\begin{array}{cc}1& -1\\ 0& 2\\ 5& 0\end{array}\right]$$$$=\left[\begin{array}{cc}2\cdot 1+1\cdot 0+3\cdot 5& 2\cdot (-1)+1\cdot 2+3\cdot 0\\ 4\cdot 1+1\cdot 0+0\cdot 5& 4\cdot (-1)+1\cdot 2+0\cdot 0\end{array}\right]$$$$=\left[\begin{array}{cc}17& 0\\ 4& -2\end{array}\right]$$$$\therefore {\left(AB\right)}^{\prime }={\left[\begin{array}{cc}17& 0\\ 4& -2\end{array}\right]}^{{}^{\prime }}=\left[\begin{array}{cc}17& 4\\ 0& -2\end{array}\right]$$$$B^{\prime }{A}^{\prime }={\left[\begin{array}{cc}1& -1\\ 0& 2\\ 5& 0\end{array}\right]}^{{}^{\prime }}\cdot {\left[\begin{array}{ccc}2& 1& 3\\ 4& 1& 0\end{array}\right]}^{{}^{\prime }}$$$$=\left[\begin{array}{ccc}1& 0& 5\\ -1& 2& 0\end{array}\right]\left[\begin{array}{cc}2& 4\\ 1& 1\\ 3& 0\end{array}\right]$$$$=\left[\begin{array}{cc}1\cdot 2+0\cdot 1+5\cdot 3& 1\cdot 4+0\cdot 1+5\cdot 0\\ (-1)\cdot 2+2\cdot 1+0\cdot 3& (-1)\cdot 4+2\cdot 1+0\cdot 0\end{array}\right]$$$$=\left[\begin{array}{cc}17& 4\\ 0& -2\end{array}\right]$$$$\mathrm{Hence},{\left(AB\right)}^{\prime }=B^{\prime }{A}^{\prime }$$

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