Cayley-Hamilton Theorem with a fraction

Question

A ∈ M3,3(R) be the following matrix:

$$A = \frac{1}{4} \left( \begin{matrix} 3 & 2 & 7 \\ -10 & 4 & -30\\ -1 & 2 & -3 \end{matrix} \right)$$

We have been asked to find out the characteristic polynomial PA(t) and to use the Cayley Hamilton Theorem to compute $A^5$ on the above matrix. how do you calculate the Characteristic Polynomial and Cayley Hamilton Theorem to a matrix with a fraction outside.

Sorry for the poor writing, I do no know how to put a fraction outside a matrix.

Answer 1

In general for an $n\times n$ matrix $M$, $\det cM = c^n\det M$.

So, let $4\lambda = \mu$ and compute the characteristic polynomial as $\det \lambda I - \frac14 A = \det \frac14 \mu I-\frac14 A = \left(\frac14\right)^3 \det \mu I - A$.

Back-substitute at the end.