find the characteristic equation of A =

2 | 2 | 0 |

2 | 1 | 1 |

-7 | 2 | -3 |

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.

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