Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACESdeterminant defined by (8.45) and their corresponding cofactors, we write |A| asthe 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 13A 32 A 33∣ ∣∣∣+ A 22∣ ∣∣∣ A 11 A 13A 31 A 33∣ ∣∣∣− A 23∣ ∣∣∣ A 11 A 12A 31 A 32∣ ∣∣∣.We will see later that the value of the determinant is independent of the rowor 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 theorder-2 determinants.◮Evaluate the determinant ∣ ∣∣∣ A 12 A 13A 32 A 33∣ ∣∣∣.By considering the products of the elements of the first row in the determinant, and theircorresponding cofactors, we find∣ A ∣12 A 13 ∣∣∣= AA 32 A 12 (−1) (1+1) |A 33 | + A 13 (−1) (1+2) |A 32 |33= A 12 A 33 − A 13 A 32 ,where the values of the order-1 determinants |A 33 | and |A 32 | are defined to be A 33 and A 32respectively. It must be remembered that the determinant is not the 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 usuallyremembered and is the form that is obtained immediately by considering theLaplace expansion using the first row of the determinant. The last equality, whichessentially rearranges a Laplace expansion using the second row into one usingthe first row, supports our assertion that the value of the determinant is unaffectedby which row or column is chosen for the expansion.260