A = left[ {begin{array}{*{20}{c}}1&0&00&1&10&{ - 2}&4end{array}} right];,,I = left[ {begin{array}{*{20}{c}}1&0&00&1&00&0&1end{array}} right] ; {A^{ - 1}} = frac{1}{6}[{A^2} + cA + dI] કે જ્યાં c,d in

Q: $A = \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&1\\0&{ - 2}&4\end{array}} \right];\,\,I = \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right] ; {A^{ - 1}} = \frac{1}{6}[{A^2} + cA + dI]$ કે જ્યાં $c,d \in R$, તો $(c,d) =\ . ....... .$

Given $A = \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&1\\0&{ - 2}&4\end{array}} \right],\,\,{A^{ - 1}} = \frac{1}{6}\left[ {\begin{array}{*{20}{c}}6&0&0\\0&4&{ - 1}\\0&2&1\end{array}} \right]$ ${A^2} = \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&1\\0&{ - 2}&4\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&1\\0&{ - 2}&4\end{array}} \right]\, $ $= \left[ {\begin{array}{*{20}{c}}1&0&0\\0&{ - 1}&5\\0&{ - 10}&{14}\end{array}} \right]$ $cA = \left[ {\begin{array}{*{20}{c}}c&0&0\\0&c&c\\0&{ - 2c}&{4c}\end{array}} \right]\,$; $dI = \left[ {\begin{array}{*{20}{c}}d&0&0\\0&d&0\\0&0&d\end{array}} \right]$ $\therefore $ By ${A^{ - 1}} = \frac{1}{6}[{A^2} + cA + dI]$ $ 6 = 1 + c + d, ($By equality of matrices$)$ $\therefore (-6,11)$ satisfy the relation.

MCQ

ગુજરાતી માધ્યમ

$A = \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&1\\0&{ - 2}&4\end{array}} \right];\,\,I = \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right] ; {A^{ - 1}} = \frac{1}{6}[{A^2} + cA + dI]$ કે જ્યાં $c,d \in R$, તો $(c,d) =\ . ....... .$

A$(6, 11)$
B$(6, -11)$
C$(-6, 11)$
D$(-6, -11)$

IIT 2005, Difficult

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

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