in any element of the matrix Adj(A−λI ).Observe that the equation Adj(A−λI )can
be written in the form
Adj(A−λI ) = B 1 λn−^1 +B 2 λn−^2 +··· +Bn− 2 λ^2 +Bn− 1 λ+Bn,
where B 1 , B 2 ,... , B nare n×nmatrices not containing λand in general depend upon
the elements of A. Using the relation
(A−λI )Adj(A−λI ) = |A−λI |I=C(λ)I
Write the right-hand side as
C(λ)I=λnI+α 1 λn−^1 I+···+αn− 2 λ^2 I+αn− 1 λI +αnI,
and write the left-hand side as
(A−λI )(B 1 λn−^1 +B 2 λn−^2 +···+Bn− 2 λ^2 +Bn− 1 λ+Bn)
=−B 1 λn−B 2 λn−^1 −.. .−Bn− 2 λ^3 −Bn− 1 λ^2 −Bnλ
+AB 1 λn−^1 +···+AB n− 3 λ^3 +AB n− 2 λ^2 +AB n− 1 λ+AB n
By comparing the left and right-hand sides of this equation one can equate the
coefficients of like powers of λand obtain the following equations.
−B 1 =I
AB 1 −B 2 =α 1 I
AB 2 −B 3 =α 2 I
..
.
AB n− 3 −Bn− 2 =αn− 3 I
AB n− 2 −Bn− 1 =αn− 2 I
AB n− 1 −Bn=αn− 1 I
AB n=αnI
: An
: An−^1
: An−^2
..
.
: A^3
: A^2
: A
: I
Now multiply the first equation by An, the second equation by An−^1 , the third
equation by An−^2 ,... , the second to last equation by Aand the last equation by I.
The multiplication factors are illustrated to the right-hand side of the equations
listed above. After multiplication, the equations are summed. Note that on the
right-hand side there results the matrix equation C(A) and on the left-hand side
