Find the characteristic polynomial of $(M^{-1})^3$

Question

Given that M is a square matrix with characteristic polynomial

$p_{m}(x) = -x^3 +6x^2+9x-14$

Find the characteristic polynomial of $(M^{-1})^3$

My attempt:

x of $(M^{-1})^3$ is $1^3$, $(-2)^3$ , $7^3$ = $1$ , $-8$ , $343$

$p_{(m^-1)^3} = (x-1)(x+8)(x-343)$
or $-(x-1)(x+8)(x-343) $

= $x^3-336x^2-2409x+2744$
or $-x^3+336x^2+2409x-2744 $

Is my approach correct?

Answer 1

Note that $-x^3+6x^2+9x-14=-(x-1)(x+2)(x-7)$, so we may assume that $M$ is the diagonal matrix with entries $1,-2,7$ on the diagonal. Then it's easy to see that the characteristic polynomial of $M^{-3}$ is given by $(x-1)(x+(1/2)^3)(x-(1/7)^3)$.