By James Dean Brown

This article presents a pragmatic, entire review of the various levels and actions fascinated by constructing and imposing a legitimate, rational, and potent language software. It systematically describes and exemplifies all of the parts of language curriculum layout. actions and routines, photograph organizers, and pattern language courses illustrate and advertise pedagogically sound perform and potent integration of fabric.

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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 ; .