1
$\begingroup$

enter image description here

I have computed (a) to be $-\lambda^3$. I also know that the Cayley-Hamilton theorem states that substituting the matrix A (where A is matrix with p(λ)=det(λI-A) for λ in this polynomial results in the zero matrix p(A)=0. But don't know how this verifies for f. I'm unsure how I find (c).

$\endgroup$
3
$\begingroup$

It is easy to see if you choose the right basis. The obvious choice is $1,x,x^2$. With respect to such basis, the matrix of $f$ is $$ \begin{bmatrix} 0&1&0\\0&0&2\\0&0&0\end{bmatrix}. $$ Then you can easily see that $f^3=0$ (which is the fact that the third derivative of a degree 3 polynomial is zero). And the characteristic polynomial is also the minimal polynomial, because the minimal polynomial divides the characteristic polynomial; but $f^2\ne0$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.