If A = [ * 20 c 0&1 1&0 ],B = [ * 20 c 0& - i &0 ] then (A + B)^2… - Vidyadip

MCQ

If $A = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right],B = \left[ {\begin{array}{*{20}{c}}0&{ - i}\\i&0\end{array}} \right]$ then ${(A + B)^2}$ equals

✓
${A^2} + {B^2}$
B
${A^2} + {B^2} + 2AB$
C
${A^2} + {B^2} + AB - BA$
D
None of these

✓

Answer

Correct option: A.

${A^2} + {B^2}$

a
(a) $AB = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}0&{ - i}\\i&0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}i&0\\0&{ - i}\end{array}} \right]$

and $BA = \left[ {\begin{array}{*{20}{c}}0&{ - i}\\i&0\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - i}&0\\0&i\end{array}} \right] = - AB$

$\therefore AB + BA = O$

Hence, ${(A + B)^2} = {A^2} + {B^2}$.

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