JEE MAIN - Mathematics (2021 - 27th July Morning Shift - No. 9)
Let $$A = \left[ {\matrix{
1 & 2 \cr
{ - 1} & 4 \cr
} } \right]$$. If A$$-$$1 = $$\alpha$$I + $$\beta$$A, $$\alpha$$, $$\beta$$ $$\in$$ R, I is a 2 $$\times$$ 2 identity matrix then 4($$\alpha$$ $$-$$ $$\beta$$) is equal to :
5
$${8 \over 3}$$
2
4
Explanation
$$A = \left[ {\matrix{
1 & 2 \cr
{ - 1} & 4 \cr
} } \right],|A| = 6$$
$${A^{ - 1}} = {{adjA} \over {|A|}} = {1 \over 6}\left[ {\matrix{ 4 & { - 2} \cr 1 & 1 \cr } } \right] = \left[ {\matrix{ {{2 \over 3}} & { - {1 \over 3}} \cr {{1 \over 6}} & {{1 \over 6}} \cr } } \right]$$
$$\left[ {\matrix{ {{2 \over 3}} & { - {1 \over 3}} \cr {{1 \over 6}} & {{1 \over 6}} \cr } } \right] = \left[ {\matrix{ \alpha & 0 \cr 0 & \alpha \cr } } \right] + \left[ {\matrix{ \beta & {2\beta } \cr { - \beta } & {4\beta } \cr } } \right]$$
$$\left. \matrix{ \alpha + \beta = {2 \over 3} \hfill \cr \beta = - {1 \over 6} \hfill \cr} \right\} \Rightarrow \alpha = {2 \over 3} + {1 \over 6} = {5 \over 6}$$
$$ \therefore $$ $$4(\alpha - \beta ) = 4(1) = 4$$
$${A^{ - 1}} = {{adjA} \over {|A|}} = {1 \over 6}\left[ {\matrix{ 4 & { - 2} \cr 1 & 1 \cr } } \right] = \left[ {\matrix{ {{2 \over 3}} & { - {1 \over 3}} \cr {{1 \over 6}} & {{1 \over 6}} \cr } } \right]$$
$$\left[ {\matrix{ {{2 \over 3}} & { - {1 \over 3}} \cr {{1 \over 6}} & {{1 \over 6}} \cr } } \right] = \left[ {\matrix{ \alpha & 0 \cr 0 & \alpha \cr } } \right] + \left[ {\matrix{ \beta & {2\beta } \cr { - \beta } & {4\beta } \cr } } \right]$$
$$\left. \matrix{ \alpha + \beta = {2 \over 3} \hfill \cr \beta = - {1 \over 6} \hfill \cr} \right\} \Rightarrow \alpha = {2 \over 3} + {1 \over 6} = {5 \over 6}$$
$$ \therefore $$ $$4(\alpha - \beta ) = 4(1) = 4$$
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