"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 :"

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"$$A = \left[ {\matrix{ 1 & 2 \cr { - 1} & 4 \cr } } \right],|A| = 6$$

$${A^{ - 1}} = {{adjA} \over {|A|}} = {1 \over 6}\left[ {\matrix{ 4 & { - 2} \cr 1 & 1 \cr } } \right] = \left[ {\matrix{ {{2 \over 3}} & { - {1 \over 3}} \cr {{1 \over 6}} & {{1 \over 6}} \cr } } \right]$$

$$\left[ {\matrix{ {{2 \over 3}} & { - {1 \over 3}} \cr {{1 \over 6}} & {{1 \over 6}} \cr } } \right] = \left[ {\matrix{ \alpha & 0 \cr 0 & \alpha \cr } } \right] + \left[ {\matrix{ \beta & {2\beta } \cr { - \beta } & {4\beta } \cr } } \right]$$

$$\left. \matrix{ \alpha + \beta = {2 \over 3} \hfill \cr \beta = - {1 \over 6} \hfill \cr} \right\} \Rightarrow \alpha = {2 \over 3} + {1 \over 6} = {5 \over 6}$$

$\therefore$ $4(\alpha - \beta ) = 4(1) = 4$"

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 :

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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 :

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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 :