MATRICES AND VECTOR SPACES
determinant defined by (8.45) and their corresponding cofactors, we write|A|as
the Laplace expansion
|A|=A 21 (−1)(2+1)M 21 +A 22 (−1)(2+2)M 22 +A 23 (−1)(2+3)M 23=−A 21∣
∣
∣
∣A 12 A 13
A 32 A 33∣
∣
∣
∣+A^22∣
∣
∣
∣A 11 A 13
A 31 A 33∣
∣
∣
∣−A^23∣
∣
∣
∣A 11 A 12
A 31 A 32∣
∣
∣
∣.We will see later that the value of the determinant is independent of the row
or column chosen. Of course, we have not yet determined the value of|A|but,
rather, written it as the weighted sum of three determinants of order 2. However,
applying again the definition of a determinant, we can evaluate each of the
order-2 determinants.
Evaluate the determinant ∣
∣
∣
∣A 12 A 13
A 32 A 33
∣∣
∣
∣.
By considering the products of the elements of the first row in the determinant, and their
corresponding cofactors, we find
∣∣
∣
∣A 12 A 13
A 32 A 33
∣∣
∣
∣=A^12 (−1)
(1+1)|A 33 |+A 13 (−1)(1+2)|A 32 |
=A 12 A 33 −A 13 A 32 ,
where the values of the order-1 determinants|A 33 |and|A 32 |are defined to beA 33 andA 32
respectively. It must be remembered that the determinant isnotthe same as the modulus,
e.g. det (−2) =|− 2 |=−2, not 2.
We can now combine all the above results to show that the value of thedeterminant (8.45) is given by
|A|=−A 21 (A 12 A 33 −A 13 A 32 )+A 22 (A 11 A 33 −A 13 A 31 )−A 23 (A 11 A 32 −A 12 A 31 ) (8.46)=A 11 (A 22 A 33 −A 23 A 32 )+A 12 (A 23 A 31 −A 21 A 33 )+A 13 (A 21 A 32 −A 22 A 31 ), (8.47)where the final expression gives the form in which the determinant is usually
remembered and is the form that is obtained immediately by considering the
Laplace expansion using the first row of the determinant. The last equality, which
essentially rearranges a Laplace expansion using the second row into one using
the first row, supports our assertion that the value of the determinant is unaffected
by which row or column is chosen for the expansion.