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If $$A = \left[ {\begin{matrix} {5a} & { - b} \\ 3 & 2 \\ \end{matrix} } \right]$$ and $$A$$ adj $$A=A$$ $${A^T},$$ then $$5a+b$$ is equal to :

JEE · Math · previous-year question

  1. A.$$4$$
  2. B.$$13$$
  3. C.$$-1$$
  4. D.$$5$$correct

Answer

D. $$5$$

Explanation

$$A\left( {Adj\,\,A} \right) = A\,{A^T}$$ $$ \Rightarrow {A^{ - 1}}A\left( {adj\,\,A} \right) = {A^{ - 1}}A\,{A^T}$$ $$Adj\,\,A = {A^T}$$ $$ \Rightarrow \left[ {\begin{matrix} 2 & b \\ { - 3} & {5a} \\ \end{matrix} } \right] = \left[ {\begin{matrix} {5a} & 3 \\ { - b} & 2 \\ \end{matrix} } \right]$$ $$ \Rightarrow a = {2 \over 5}\,\,$$ and $$\,\,b = 3$$ $$ \Rightarrow 5a + b = 5$$

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