Let A = \left[ {\begin{matrix} 1 & 2 \ { - 1} & 4 \ \end{matrix} } ight]. If A-1 = \alphaI + \betaA, \alpha, \beta \in R, I is a 2 imes 2 identity matrix then 4(\alpha - \beta) is equal to :

Question

Let A = \left[ {\begin{matrix} 1 & 2 \ { - 1} & 4 \ \end{matrix} } ight]. If A-1 = \alphaI + \betaA, \alpha, \beta \in R, I is a 2 	imes 2 identity matrix then 4(\alpha - \beta) is equal to :

Accepted Answer

Answer: D. 4 A = \left[ {\begin{matrix} 1 & 2 \ { - 1} & 4 \ \end{matrix} } ight],|A| = 6 {A^{ - 1}} = {{adjA} \over {|A|}} = {1 \over 6}\left[ {\begin{matrix} 4 & { - 2} \ 1 & 1 \ \end{matrix} } ight] = \left[ {\begin{matrix} {{2 \over 3}} & { - {1 \over 3}} \ {{1 \over 6}} & {{1 \over 6}} \ \end{matrix} } ight] \left[ {\begin{matrix} {{2 \over 3}} & { - {1 \over 3}} \ {{1 \over 6}} & {{1 \over 6}} \ \end{matrix} } ight] = \left[ {\begin{matrix} \alpha & 0 \ 0 & \alpha \ \end{matrix} } ight] + \left[ {\begin{matrix} \beta & {2\beta } \ { - \beta } & {4\beta } \ \end{matrix} } ight] \left. \begin{matrix} \alpha + \beta = {2 \over 3} \hfill \ \beta = - {1 \over 6} \hfill \\end{matrix} ight\} \Rightarrow \alpha = {2 \over 3} + {1 \over 6} = {5 \over 6} 	herefore 4(\alpha - \beta ) = 4(1) = 4

Let $$A = \left[ {\begin{matrix} 1 & 2 \\ { - 1} & 4 \\ \end{matrix} } \right]$$. 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 :

Answer

Explanation

More JEE Math questions