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3 and 4 .Determinants and Matrices
easy
જો $A = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right],B = \left[ {\begin{array}{*{20}{c}}0&{ - i}\\i&0\end{array}} \right]$ તો ${(A + B)^2}$ = . . .
A
${A^2} + {B^2}$
B
${A^2} + {B^2} + 2AB$
C
${A^2} + {B^2} + AB - BA$
D
એકપણ નહી.
Solution
(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, ${(A + B)^2} = {A^2} + {B^2}$.
Standard 12
Mathematics
