Mathematical Methods for Physics and Engineering - Matematica.NET

8.9 THE DETERMINANT OF A MATRIXwhich shows that the trace of a multiple product is invariant under cyclicpermutations of the matrices in the product. Other easily derived properties ofthe trace are, for example, Tr A T =TrA and Tr A † =(TrA) ∗ .8.9 The determinant of a matrixFor a given matrix A, the determinant det A (like the trace) is a single number (oralgebraic expression) that depends upon the elements of A. Also like the trace,the determinant is defined only for square matrices. If, for example, A is a 3 × 3matrix then its determinant, of order 3, is denoted bydet A = |A| =∣A 11 A 12 A 13A 21 A 22 A 23A 31 A 32 A 33∣ ∣∣∣∣∣. (8.45)In order to calculate the value of a determinant, we first need to introducethe notions of the minor and the cofactor of an element of a matrix. (Weshall see that we can use the cofactors to write an order-3 determinant as theweighted sum of three order-2 determinants, thereby simplifying its evaluation.)The minor M ij of the element A ij of an N × N matrix A is the determinant ofthe (N − 1) × (N − 1) matrix obtained by removing all the elements of the ithrow and jth column of A; the associated cofactor, C ij , is found by multiplyingthe minor by (−1) i+j .◮Find the cofactor of the element A 23 of the matrix⎛A = ⎝ A ⎞11 A 12 A 13A 21 A 22 A 23⎠ .A 31 A 32 A 33Removing all the elements of the second row and third column of A and forming thedeterminant of the remaining terms gives the minorM 23 =∣ A ∣11 A 12 ∣∣∣.A 31 A 32Multiplying the minor by (−1) 2+3 =(−1) 5 = −1 givesC 23 = −∣ A ∣11 A 12 ∣∣∣. ◭A 31 A 32We now define a determinant as the sum of the products of the elements of anyrow or column and their corresponding cofactors, e.g.A 21 C 21 + A 22 C 22 + A 23 C 23 orA 13 C 13 + A 23 C 23 + A 33 C 33 . Such a sum is called a Laplace expansion. For example,in the first of these expansions, using the elements of the second row of the259