JEE MAIN - Mathematics (2016 (Offline) - No. 10)
If $$A = \left[ {\matrix{
{5a} & { - b} \cr
3 & 2 \cr
} } \right]$$ and $$A$$ adj $$A=A$$ $${A^T},$$ then $$5a+b$$ is equal to :
$$4$$
$$13$$
$$-1$$
$$5$$
Explanation
$$A\left( {Adj\,\,A} \right) = A\,{A^T}$$
$$ \Rightarrow {A^{ - 1}}A\left( {adj\,\,A} \right) = {A^{ - 1}}A\,{A^T}$$
$$Adj\,\,A = {A^T}$$
$$ \Rightarrow \left[ {\matrix{ 2 & b \cr { - 3} & {5a} \cr } } \right] = \left[ {\matrix{ {5a} & 3 \cr { - b} & 2 \cr } } \right]$$
$$ \Rightarrow a = {2 \over 5}\,\,$$ and $$\,\,b = 3$$
$$ \Rightarrow 5a + b = 5$$
$$ \Rightarrow {A^{ - 1}}A\left( {adj\,\,A} \right) = {A^{ - 1}}A\,{A^T}$$
$$Adj\,\,A = {A^T}$$
$$ \Rightarrow \left[ {\matrix{ 2 & b \cr { - 3} & {5a} \cr } } \right] = \left[ {\matrix{ {5a} & 3 \cr { - b} & 2 \cr } } \right]$$
$$ \Rightarrow a = {2 \over 5}\,\,$$ and $$\,\,b = 3$$
$$ \Rightarrow 5a + b = 5$$
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