Computing the characteristic polynomial

Question

Consider the following matrix A over the field $F_7$ $$ \left(\begin{array}{rrr} 3 & 4 & 4 \\ 2 & 5 & 2 \\ 1 & 2 & 5 \end{array}\right) . $$

I'm asked to compute the characteristic polynomial χA of A

I understand that I need to find the characteristic equation of $\det(A- \lambda I)$ but do I need to work in $F_7$ from the start or can I change it at the end

not in field 7 I believe I got $-x^3+13x^2-39x+27$

can I change it to this $6x^3+6x^2+3x+6$ ? would this be correct?

Answer 1

What you have done is correct (assuming the $x$s in your final answer should be $\lambda$s). But two points.

• I would say there is no need to avoid negatives in describing $F_7$, so your answer can be simplified to $-\lambda^3-\lambda^2+3\lambda-1$. (However your instructor may disagree - better check.)
• IMHO it would actually be easier if you used $F_7$ from the start, reducing the numbers to "least absolute value" form: $$\eqalign{\det(A-\lambda I) &=\det\pmatrix{3-\lambda&-3&-3\cr 2&-2-\lambda&2\cr 1&2&-2-\lambda\cr}\cr &=(3-\lambda)(-3\lambda+\lambda^2)+3(1-2\lambda)-3(-1+\lambda)\cr &=-\lambda^3-\lambda^2+3\lambda-1\ .\cr}$$