"If the matrix $A = \left( {\matrix{ 0 & 2 \cr K & { - 1} \cr } } \right)$ satisfies $A({A^3} + 3I) = 2I$, then the value of K is :"

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"Given matrix $A = \left[ {\matrix{ 0 & 2 \cr k & { - 1} \cr } } \right]$

${A^4} + 3IA = 2I$

$\Rightarrow {A^4} = 2I - 3A$

Also characteristic equation of A is $|A - \lambda I|\, = 0$

$$ \Rightarrow \left| {\matrix{ {0 - \lambda } & 2 \cr k & { - 1 - \lambda } \cr } } \right| = 0$$

$\Rightarrow \lambda + {\lambda ^2} - 2k = 0$

$\Rightarrow A + {A^2} = 2K.I$

$\Rightarrow {A^2} = 2KI - A$

$\Rightarrow {A^4} = 4{K^2}I + {A^2} - 4AK$

Put ${A^2} = 2KI - A$

and ${A^4} = 2I - 3A$

$2I - 3A = 4{K^2}I + 2KI - A - 4AK$

$\Rightarrow I(2 - 2K - 4{K^2}) = A(2 - 4K)$

$\Rightarrow - 2I(2{K^2} + K - 1) = 2A(1 - 2K)$

$\Rightarrow - 2I(2K - 1)(K + 1) = 2A(1 - 2K)$

$\Rightarrow (2K - 1)(2A) - 2I(2K - 1)(K + 1) = 0$

$\Rightarrow (2K - 1)[2A - 2I(K + 1)] = 0$

$\Rightarrow K = {1 \over 2}$"

If the matrix $A = \left( {\matrix{ 0 & 2 \cr K & { - 1} \cr } } \right)$ satisfies $A({A^3} + 3I) = 2I$, then the value of K is :

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If the matrix $A = \left( {\matrix{ 0 & 2 \cr K & { - 1} \cr } } \right)$ satisfies $A({A^3} + 3I) = 2I$, then the value of K is :

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