Question Number 164134 by Avijit007 last updated on 14/Jan/22
Answered by Rasheed.Sindhi last updated on 14/Jan/22
![A= [(a_(11) ,a_(12) ),(a_(21) ,a_(22) ) ],B= [(b_(11) ,b_(12) ),(b_(21) ,b_(22) ) ](say) •2A+B^t = [((2a_(11) ),(2a_(12) )),((2a_(21) ),(2a_(22) )) ]+ [(b_(11) ,b_(21) ),(b_(12) ,b_(22) ) ]= [(2,5),((10),2) ] [((2a_(11) +b_(11) ),(2a_(12) +b_(21) )),((2a_(21) +b_(12) ),(2a_(22) +b_(22) )) ]= [(2,5),((10),2) ] •2B+A^t = [((2b_(11) ),(2b_(12) )),((2b_(21) ),(2b_(22) )) ]+ [(a_(11) ,a_(21) ),(a_(12) ,a_(22) ) ]= [(1,8),(4,1) ] [((2b_(11) +a_(11) ),(2b_(12) +a_(21) )),((2b_(21) +a_(12) ),(2b_(22) +a_(22) )) ]= [(1,8),(4,1) ] { ((2a_(11) +b_(11) =2⇒2a_(11) +b_(11) =2)),((2b_(11) +a_(11) =1⇒2a_(11) +4b_(11) =2)) :} ⇒3b_(11) =0⇒b_(11) =0⇒a_(11) =1 { ((2a_(22) +b_(22) =2⇒2a_(22) +b_(22) =2)),((2b_(22) +a_(22) =1⇒2a_(22) +4b_(22) =2)) :} ⇒3b_(22) =0⇒b_(22) =0⇒a_(22) =1 { ((2a_(12) +b_(21) =5⇒2a_(12) +b_(21) =5)),((2b_(21) +a_(12) =4⇒2a_(12) +4b_(21) =8)) :} 3b_(21) =3⇒b_(21) =1⇒a_(12) =2 { ((2a_(21) +b_(12) =10⇒2a_(21) +b_(12) =10)),((2b_(12) +a_(21) =8⇒2a_(21) +4b_(12) =16)) :} 3b_(12) =6⇒b_(12) =2⇒a_(21) =4 A= [(a_(11) ,a_(12) ),(a_(21) ,a_(22) ) ]= [(1,2),(4,1) ] B= [(b_(11) ,b_(12) ),(b_(21) ,b_(22) ) ]= [(0,2),(1,0) ]](https://www.tinkutara.com/question/Q164153.png)
$${A}=\begin{bmatrix}{{a}_{\mathrm{11}} }&{{a}_{\mathrm{12}} }\\{{a}_{\mathrm{21}} }&{{a}_{\mathrm{22}} }\end{bmatrix},{B}=\begin{bmatrix}{{b}_{\mathrm{11}} }&{{b}_{\mathrm{12}} }\\{{b}_{\mathrm{21}} }&{{b}_{\mathrm{22}} }\end{bmatrix}\left({say}\right) \\ $$$$\bullet\mathrm{2}{A}+{B}^{{t}} =\begin{bmatrix}{\mathrm{2}{a}_{\mathrm{11}} }&{\mathrm{2}{a}_{\mathrm{12}} }\\{\mathrm{2}{a}_{\mathrm{21}} }&{\mathrm{2}{a}_{\mathrm{22}} }\end{bmatrix}+\begin{bmatrix}{{b}_{\mathrm{11}} }&{{b}_{\mathrm{21}} }\\{{b}_{\mathrm{12}} }&{{b}_{\mathrm{22}} }\end{bmatrix}=\begin{bmatrix}{\mathrm{2}}&{\mathrm{5}}\\{\mathrm{10}}&{\mathrm{2}}\end{bmatrix} \\ $$$$\begin{bmatrix}{\mathrm{2}{a}_{\mathrm{11}} +{b}_{\mathrm{11}} }&{\mathrm{2}{a}_{\mathrm{12}} +{b}_{\mathrm{21}} }\\{\mathrm{2}{a}_{\mathrm{21}} +{b}_{\mathrm{12}} }&{\mathrm{2}{a}_{\mathrm{22}} +{b}_{\mathrm{22}} }\end{bmatrix}=\begin{bmatrix}{\mathrm{2}}&{\mathrm{5}}\\{\mathrm{10}}&{\mathrm{2}}\end{bmatrix} \\ $$$$\bullet\mathrm{2}{B}+{A}^{{t}} =\begin{bmatrix}{\mathrm{2}{b}_{\mathrm{11}} }&{\mathrm{2}{b}_{\mathrm{12}} }\\{\mathrm{2}{b}_{\mathrm{21}} }&{\mathrm{2}{b}_{\mathrm{22}} }\end{bmatrix}+\begin{bmatrix}{{a}_{\mathrm{11}} }&{{a}_{\mathrm{21}} }\\{{a}_{\mathrm{12}} }&{{a}_{\mathrm{22}} }\end{bmatrix}=\begin{bmatrix}{\mathrm{1}}&{\mathrm{8}}\\{\mathrm{4}}&{\mathrm{1}}\end{bmatrix} \\ $$$$\begin{bmatrix}{\mathrm{2}{b}_{\mathrm{11}} +{a}_{\mathrm{11}} }&{\mathrm{2}{b}_{\mathrm{12}} +{a}_{\mathrm{21}} }\\{\mathrm{2}{b}_{\mathrm{21}} +{a}_{\mathrm{12}} }&{\mathrm{2}{b}_{\mathrm{22}} +{a}_{\mathrm{22}} }\end{bmatrix}=\begin{bmatrix}{\mathrm{1}}&{\mathrm{8}}\\{\mathrm{4}}&{\mathrm{1}}\end{bmatrix} \\ $$$$\begin{cases}{\mathrm{2}{a}_{\mathrm{11}} +{b}_{\mathrm{11}} =\mathrm{2}\Rightarrow\mathrm{2}{a}_{\mathrm{11}} +{b}_{\mathrm{11}} =\mathrm{2}}\\{\mathrm{2}{b}_{\mathrm{11}} +{a}_{\mathrm{11}} =\mathrm{1}\Rightarrow\mathrm{2}{a}_{\mathrm{11}} +\mathrm{4}{b}_{\mathrm{11}} =\mathrm{2}}\end{cases} \\ $$$$\Rightarrow\mathrm{3}{b}_{\mathrm{11}} =\mathrm{0}\Rightarrow{b}_{\mathrm{11}} =\mathrm{0}\Rightarrow{a}_{\mathrm{11}} =\mathrm{1} \\ $$$$\begin{cases}{\mathrm{2}{a}_{\mathrm{22}} +{b}_{\mathrm{22}} =\mathrm{2}\Rightarrow\mathrm{2}{a}_{\mathrm{22}} +{b}_{\mathrm{22}} =\mathrm{2}}\\{\mathrm{2}{b}_{\mathrm{22}} +{a}_{\mathrm{22}} =\mathrm{1}\Rightarrow\mathrm{2}{a}_{\mathrm{22}} +\mathrm{4}{b}_{\mathrm{22}} =\mathrm{2}}\end{cases} \\ $$$$\Rightarrow\mathrm{3}{b}_{\mathrm{22}} =\mathrm{0}\Rightarrow{b}_{\mathrm{22}} =\mathrm{0}\Rightarrow{a}_{\mathrm{22}} =\mathrm{1} \\ $$$$\begin{cases}{\mathrm{2}{a}_{\mathrm{12}} +{b}_{\mathrm{21}} =\mathrm{5}\Rightarrow\mathrm{2}{a}_{\mathrm{12}} +{b}_{\mathrm{21}} =\mathrm{5}}\\{\mathrm{2}{b}_{\mathrm{21}} +{a}_{\mathrm{12}} =\mathrm{4}\Rightarrow\mathrm{2}{a}_{\mathrm{12}} +\mathrm{4}{b}_{\mathrm{21}} =\mathrm{8}}\end{cases} \\ $$$$\mathrm{3}{b}_{\mathrm{21}} =\mathrm{3}\Rightarrow{b}_{\mathrm{21}} =\mathrm{1}\Rightarrow{a}_{\mathrm{12}} =\mathrm{2} \\ $$$$\begin{cases}{\mathrm{2}{a}_{\mathrm{21}} +{b}_{\mathrm{12}} =\mathrm{10}\Rightarrow\mathrm{2}{a}_{\mathrm{21}} +{b}_{\mathrm{12}} =\mathrm{10}}\\{\mathrm{2}{b}_{\mathrm{12}} +{a}_{\mathrm{21}} =\mathrm{8}\Rightarrow\mathrm{2}{a}_{\mathrm{21}} +\mathrm{4}{b}_{\mathrm{12}} =\mathrm{16}}\end{cases} \\ $$$$\mathrm{3}{b}_{\mathrm{12}} =\mathrm{6}\Rightarrow{b}_{\mathrm{12}} =\mathrm{2}\Rightarrow{a}_{\mathrm{21}} =\mathrm{4} \\ $$$${A}=\begin{bmatrix}{{a}_{\mathrm{11}} }&{{a}_{\mathrm{12}} }\\{{a}_{\mathrm{21}} }&{{a}_{\mathrm{22}} }\end{bmatrix}=\begin{bmatrix}{\mathrm{1}}&{\mathrm{2}}\\{\mathrm{4}}&{\mathrm{1}}\end{bmatrix} \\ $$$${B}=\begin{bmatrix}{{b}_{\mathrm{11}} }&{{b}_{\mathrm{12}} }\\{{b}_{\mathrm{21}} }&{{b}_{\mathrm{22}} }\end{bmatrix}=\begin{bmatrix}{\mathrm{0}}&{\mathrm{2}}\\{\mathrm{1}}&{\mathrm{0}}\end{bmatrix} \\ $$