If the matrix A=pmatrix0&2&-1pmatrix satisfies A(A^3+3I)=2I, then the value of K is

Question

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

Accepted Answer

The characteristic equation of $A$ is $\lambda^2+\lambda-2K=0$, so $A^2=2KI-A$ (Cayley--Hamilton).

$A^4=(2KI-A)^2=4K^2I+A^2-4KA=4K^2I+(2KI-A)-4KA=(4K^2+2K)I-(1+4K)A.$

Also $A(A^3+3I)=2I\Rightarrow A^4=2I-3A$.

Comparing: $1+4K=3\Rightarrow K=\dfrac12$.

Solutions · 1