36 3. Intermediate Determinant Theory
=
∑
r
frA
rj
δri+
∑
s
gsA
is
δsj
=fiA
ij
+gjA
ij
which proves (D). The proof of (C) is similar but simpler.
Exercises
Prove that
1.
n
∑
r=1
n
∑
s=1
[r
k
−(r−1)
k
+s
k
−(s−1)
k
]arsA
rs
=2n
k
.
2.a
′
ij
=−
n
∑
r=1
n
∑
s=1
aisarj(A
rs
)
′
.
3.
n
∑
r=1
n
∑
s=1
(fr+gs)aisarjA
rs
=(fi+gj)aij.
Note that (2) and (3) can be obtained formally from (B) and (D), respec-
tively, by interchanging the symbolsaandAand either raising or lowering
all their parameters.
3.5 The Adjoint Determinant...................
3.5.1 Definition........................
The adjoint of a matrixA=[aij]nis denoted by adjAand is defined by
adjA=[Aji]n.
The adjoint or adjugate or a determinantA=|aij|n= detAis denoted by
adjAand is defined by
adjA=|Aji|n=|Aij|n
= det(adjA). (3.5.1)
3.5.2 The Cauchy Identity
The following theorem due to Cauchy is valid for all determinants.
Theorem.
adjA=A
n− 1
.
The proof is similar to that of the matrix relation
AadjA=AI.