જો A = left[ {begin{array}{*{20}{c}}1&{ - 1}2&{ - 1}end{array}} right],,,B = left[ {begin{array}{*{20}{c}}a&1b&{ - 1}end{array}} right] અને {(A + B)^2} = {A^2} + {B^2}, તો

Q: જો $A = \left[ {\begin{array}{*{20}{c}}1&{ - 1}\\2&{ - 1}\end{array}} \right],\,\,B = \left[ {\begin{array}{*{20}{c}}a&1\\b&{ - 1}\end{array}} \right]$ અને ${(A + B)^2} = {A^2} + {B^2}$, તો $a$ અને $b$ ની કિમતો મેળવો.

Given, $A = \left[ {\begin{array}{*{20}{c}}1&{ - 1}\\2&{ - 1}\end{array}} \right],\,\,B = \left[ {\begin{array}{*{20}{c}}a&1\\b&{ - 1}\end{array}} \right]$ $\Rightarrow A + B = \left[ {\begin{array}{*{20}{c}}{1 + a}&0\\{2 + b}&{ - 2}\end{array}} \right]$ ${A^2} = \left[ {\begin{array}{*{20}{c}}1&{ - 1}\\2&{ - 1}\end{array}} \right]\,\,\left[ {\begin{array}{*{20}{c}}1&{ - 1}\\2&{ - 1}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 1}&0\\0&{ - 1}\end{array}} \right]$ ${B^2} = \left[ {\begin{array}{*{20}{c}}a&1\\b&{ - 1}\end{array}} \right]\,\,\,\left[ {\begin{array}{*{20}{c}}a&1\\b&{ - 1}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{a^2} + b}&{a - 1}\\{ab - b}&{b + 1}\end{array}} \right]$ Also, ${A^2} + {B^2} = \left[ {\begin{array}{*{20}{c}}{{a^2} + b - 1}&{a - 1}\\{ab - b}&b\end{array}} \right]$ $\Rightarrow {(A + B)^2} = \left[ {\begin{array}{*{20}{c}}{1 + a}&0\\{2 + b}&{ - 2}\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}{1 + a}&0\\{2 + b}&{ - 2}\end{array}} \right]$ Also, ${(A + B)^2} = \left[ {\begin{array}{*{20}{c}}{{{(1 + a)}^2}}&{\,\,\,\,0}\\{(2 + b)(1 + a) - 2(2 + b)}&{\,\,\,\,4}\end{array}} \right]$ Also, ${(A + B)^2} = {A^2} + {B^2}$ $\Rightarrow \left[ {\begin{array}{*{20}{c}}{{{(1 + a)}^2}}&0\\{\,(2 + b)(a - 1)}&4\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{a^2} + b - 1}&{a - 1}\\{ab - b}&b\end{array}} \right]$ By equating, $a - 1 = 0 $ $\Rightarrow a = 1$ and $b = 4$.

MCQ

ગુજરાતી માધ્યમ

જો $A = \left[ {\begin{array}{*{20}{c}}1&{ - 1}\\2&{ - 1}\end{array}} \right],\,\,B = \left[ {\begin{array}{*{20}{c}}a&1\\b&{ - 1}\end{array}} \right]$ અને ${(A + B)^2} = {A^2} + {B^2}$, તો $a$ અને $b$ ની કિમતો મેળવો.

A$a = 4,b = 1$
B$a = 1,b = 4$
C$a = 0,b = 4$
D$a = 2,b = 4$

Medium

ધોરણ 12 સાયન્સ
ગણિત
3 and 4 . determinant and metrices
JEE

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