The Chemistry Maths Book, Second Edition

470 Chapter 16Vectors We consider here only the special case of a vector field that is the gradient of a scalar field,v 1 = 1 ∇f ...

16.11 Exercises 471 6 A detailed treatment of n-dimensional vector spaces was given by Hermann Günther Grassmann (1809–1877), Ge ...

472 Chapter 16Vectors Fora 1 = 1 (1, 2, 3),b 1 = 1 (−2, 3, −4),c 1 = 1 (0, 4, −1), find 6.a 1 + 1 b, b 1 + 1 a 7. 3 a, −a, a 2 3 ...

16.11 Exercises 473 37.Show thata 1 z 1 bis orthogonal to aand b. 38.The quantitya 1 · 1 (b 1 z 1 c)is called a triple scalar pr ...

17 Determinants 17.1 Concepts Many problems in the physical sciences, in engineering, and in statistics give rise to systems of ...

17.1 Concepts 475 Whena 1 b 2 1 − 1 b 1 a 2 is not zero, the required (unique) values of xand yare (17.4) The quantity in the de ...

476 Chapter 17Determinants The determinant (17.5) is a property of a square array of 41 = 12 2 elements, the coefficients of xan ...

17.2 Determinants of order 3 477 and, expanding the second-order determinants, D 1 = 1 a 11 a 22 a 33 1 − 1 a 11 a 23 a 32 1 − 1 ...

478 Chapter 17Determinants Minors and cofactors The minorM ij of elementa ij of a determinant Dis the determinant obtained by de ...

17.2 Determinants of order 3 479 The minorM 21 is Similarly, The complete expansion of the determinant in terms of its elements ...

480 Chapter 17Determinants The cofactorC ij of elementa ij is the minorM ij multiplied by the appropriate sign, C ij 1 = 1 (−1) ...

17.3 The general case 481 17.3 The general case A determinant of order nis a property of a square array ofn 2 elements, written ...

482 Chapter 17Determinants Each of the third-order determinants can be evaluated by the method described in Section 17.2; that i ...

17.4 The solution of linear equations 483 NOTE: on the evaluation of determinants and the solution of linear equations. The expa ...

484 Chapter 17Determinants This result can be derived by a generalization of the method described in Section 17.2 for the system ...

17.4 The solution of linear equations 485 The sum of (2) and (3) is now equal to twice (1), but this means that equation (1) con ...

486 Chapter 17Determinants The general condition that a determinant be zero is discussed in Section 17.5. Systems of linear equa ...

17.4 The solution of linear equations 487 EXAMPLE 17.8Find the values of λfor which the following system of equations has nonzer ...

488 Chapter 17Determinants The equations have nonzero solution when the determinant of the coefficients ofc 1 , c 2 ,andc 3 is z ...

17.5 Properties of determinants 489 (17.40) Multiplication by a constant If all the elements of any row (or column) are multip ...