The characteristic equation of A is A-xI=0, where x is the variable in the equation and I is the identity matrix.
We need the determinant of A-xI.
| 2-x 2 0 |
| 2 1-x 1 |
| -7 2 -3-x |=0
(2-x)(-(1-x)(3+x)-2)-2(-2(3+x)+7)=
(2-x)(-5+2x+x²)+4(3+x)-14=
-10+9x-x³+12+4x-14=
-12+13x-x³=0 or x³-13x+12=0=(x-1)(x+4)(x-3). (Eigenvalues are 1, -4, 3.) The characteristic equation is x³-13x+12=0.