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3 and 4 . determinant and metrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths3 and 4 . determinant and metrices1 Mark
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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