20.2. THE DETERMINANT 443

Sayq(λ ) = a0 +a1λ + · · ·+λ

n

Note that each entry in C (λ ) is a polynomial in λ having degree no more than n− 1. Forexample, you might have something like

C (λ ) =

 λ2 −6λ +9 3−λ 02λ −6 λ

2 −3λ 0λ −1 λ −1 λ

2 −3λ +2



=

 9 3 0−6 0 0−1 −1 2

+λ

 −6 −1 02 −3 01 1 −3

+λ2

 1 0 00 1 00 0 1

Therefore, collecting the terms in the general case,

C (λ ) =C0 +C1λ + · · ·+Cn−1λn−1

for C j some n×n matrix. Then

C (λ )(λ I −A) =(

C0 +C1λ + · · ·+Cn−1λn−1)(λ I −A) = q(λ ) I

Then multiplying out the middle term, it follows that for all |λ | sufficiently large,

a0I +a1Iλ + · · ·+ Iλn =C0λ +C1λ

2 + · · ·+Cn−1λn

−[C0A+C1Aλ + · · ·+Cn−1Aλ

n−1]

=−C0A+(C0 −C1A)λ +(C1 −C2A)λ2 + · · ·+(Cn−2 −Cn−1A)λ

n−1 +Cn−1λn

Then, using Corollary 20.2.22, one can replace λ on both sides with A. Then the right sideis seen to equal 0. Hence the left side, q(A) I is also equal to 0.

20.2.11 Cramer’s Rule

In case you are solving a system of equations, Ax= y for x, it follows that if A−1 exists,

x=(A−1A

)x= A−1 (Ax) = A−1y

thus solving the system. Now in the case that A−1 exists, there is a formula for A−1 givenabove. Using this formula,

xi =n

∑j=1

a−1i j y j =

n

∑j=1

1det(A)

cof(A) ji y j.

By the formula for the expansion of a determinant along a column,

xi =1

det(A)det

 ∗ · · · y1 · · · ∗...

......

∗ · · · yn · · · ∗

 ,

where here the ith column of A is replaced with the column vector (y1 · · · ,yn)T , and the

determinant of this modified matrix is taken and divided by det(A). This formula is knownas Cramer’s rule.