If A = [ * 20 c i&0 0&i ] and B = [ * 20 c 0& - i - i &0 ], then … - Vidyadip

MCQ

If $A = \left[ {\begin{array}{*{20}{c}}i&0\\0&i\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}0&{ - i}\\{ - i}&0\end{array}} \right]$, then $(A + B)(A - B)$ is equal to

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

✓

Answer

Correct option: A.

${A^2} - {B^2}$

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

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

Since $AB = BA,$ therefore $(A + B)(A - B) = {A^2} - {B^2}$.

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