question_answer

If A and B are square matrices such that \[B=-{{A}^{-1}}BA,\] Then find the value of \[{{(A+B)}^{2}}.\]

Answer:

We have, \[B=-\,{{A}^{-1}}BA\] \[\Rightarrow \] \[AB=-A({{A}^{-1}}BA)=-\,(A{{A}^{-1}})(BA)\] \[=-\,(l)(BA)=-\,BA\] \[\Rightarrow \] \[AB+BA=0\] Now, \[{{(A+B)}^{2}}=(A+B)(A+B)={{A}^{2}}+AB+BA+{{B}^{2}}\]             \[={{A}^{2}}+0+{{B}^{2}}\]              \[[\because \,\,\,AB=-AB]\]       \[={{A}^{2}}+{{B}^{2}}\]