16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 5.3.8

Determinants

Mathematics — Linear Algebra · characteristic-polynomial

Problem

Let

A

be a

3 \times 3

matrix over the field of complex numbers. We form the matrix

xI - A

with polynomial entries, the

i,j

entry of this matrix being the polynomial

\delta_{ij}x - A_{ij}

. If

f = \det(xI - A)

, show that

f

is a monic polynomial of degree 3. If we write

f = (x - c_1)(x - c_2)(x - c_3)

with complex numbers

c_1

c_2

, and

c_3

, prove that

c_1 + c_2 + c_3 = \operatorname{trace}(A) \quad \text{and} \quad c_1 c_2 c_3 = \det A.