If the matrix [A = ( pmatrix 0 & 2 K & - 1 pmatrix )] satisfies A(A^3 + 3I) = 2I, then the…

Question

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

Accepted Answer

Given matrix $$A = \left[ {\begin{pmatrix} 0 & 2  \
   k & { - 1}  \

\end{pmatrix} } \right]$$

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

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

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

$$ \Rightarrow \left| {\begin{pmatrix} {0 - \lambda } & 2  \
   k & { - 1 - \lambda }  \

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

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

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

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

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

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

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

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

Solutions · 1