Handbook of Mathematical Formulas by Hans-Jochen Bartsch (Auth.)

By Hans-Jochen Bartsch (Auth.)

A compilation of formulation observed through short factors and examples rang from the elemental ideas of mathematics to Fourier's sequence and the basics likelihood calculus

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65 Determinants Δχ«2 χ*3 = — , Δ α 3η Δχο where etc. e. £ = { 3 ; - 2 ; 1 } 5 Bartsch, Mathematical Formulas Δζ •• -2 -1 7 -3 17 -2 12 1. e. if all numerator determinants Δχι disappear. If Δ Φ 0 , there will be only the trivial solution Xl = X% = #3 = · · · = x n = 0 If Δ = 0 , there will be systems of solutions where n o t all variables will disappear. (b) Inhomogeneous system of equations A system of equations is called inhomogeneous if a t least one Cj Φ 0. If Δ Φ 0, the solution will be uniquely determined.

Determinant from a square matrix det A' = Arithmetic |A| det A' = det A det (AB) = det A det Β Singular matrix ( A square) If det A = Ο, A is called singular. If AB = Ν and Α φ Ν as well as Β φ Ν , then A and Β are singular. Schematically η ρ ι[Τ]-η|"ΪΓ| = m AB m,w-matrix A multiplied b y w,^-matrix Β results in the m,p-matrix C. 0 + 3-3 + 2 . 1 \ ~~ \ 2 - 1 + 4 - 2 + 1 4 2-0 + 4-3 + 1 . 1 / The commutative law does not hold generally AB Φ ΒΑ This means that, in multiplying a matrix Β by a matrix A from the left (front), the result obtained in general is different from t h a t obtained when proceeding from the right (rear).

4 5 8 13 20 . . 1 3 5 Ί ... 2 2 2 ... 5 ... First difference-sequence Second difference-sequence Third difference-sequence Fourth difference-sequence constant The primary sequence is a sequence of the fourth order. 4* 52 1. 3. ; a1 initial term an nth term (final term) For for for for q> 0 < q < q= η number of terms q quotient (ratio) 1 the sequence is increasing; q < 1 the sequence is decreasing ; 0 the sequence is oscillatory (alternating signs) ; 1 the sequence contains equal terms. 8. Sequences and series 53 Law of formation of the geometric sequence 1 ak = a^q*- for k 6 {1 ; 2 ; 3 ; .

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