Page 28 - Determinants and Their Applications in Mathematical Physics

2.3 Elementary Formulas 13

The elements come from row i of A, but the cofactors belong to the elements
in row k and are said to be alien to the elements. The identity is merely
an expansion by elements from row k of the determinant in which row k =
row i and which is therefore zero.
The identity can be combined with the expansion formula for A with the
aid of the Kronecker delta function δ ik (Appendix A.1) to form a single
identity which may be called the sum formula for elements and cofactors:
n

a ij A kj = δ ik A, 1 ≤ i ≤ n, 1 ≤ k ≤ n. (2.3.12)
j=1
It follows that
n

A ij C j =[0 ... 0 A 0 ... 0] , 1 ≤ i ≤ n,
T
j=1
where the element A is in row i of the column vector and all the other
elements are zero. If A = 0, then
n

A ij C j =0, 1 ≤ i ≤ n, (2.3.13)
j=1
that is, the columns are linearly dependent. Conversely, if the columns are
linearly dependent, then A =0.

2.3.5 Cramer’s Formula
The set of equations

n

a ij x j = b i , 1 ≤ i ≤ n,
j=1
can be expressed in column vector notation as follows:

n

C j x j = B,
j=1
where

.
T
B = b 1 b 2 b 3 ··· b n
If A = |a ij | n = 0, then the unique solution of the equations can also be
expressed in column vector notation. Let

A = C 1 C 2 ··· C j ··· C n .

Then
1
x j =
A C 1 C 2 ··· C j−1 BC j+1 ··· C n