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These determinants are either 0 (by property 9) or else ±1 (by properties 1 and 12 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of ...
Apart from this, from Class 12 civics book Politics in Indian since Independence, chapters like Rise of popular movements, ‘Era of one-party dominance’ have been removed. From the class 10 Democratic Politics-II textbooks, chapters like ‘Democracy and diversity’, ‘Popular struggles and movement’, ‘Challenges to democracy’ have ...
In the 2×2 case, if the coefficient determinant is zero, then the system is incompatible if the numerator determinants are nonzero, or indeterminate if the numerator determinants are zero. For 3×3 or higher systems, the only thing one can say when the coefficient determinant equals zero is that if any of the numerator determinants are nonzero ...
The textbooks are in color-print and are among the least expensive books in Indian book stores. [11] Textbooks created by private publishers are priced higher than those of NCERT. [ 11 ] According to a government policy decision in 2017, the NCERT will have the exclusive task of publishing central textbooks from 2018, and the role of CBSE will ...
The determinant is ρ 2 sin φ. Since dV = dx dy dz is the volume for a rectangular differential volume element (because the volume of a rectangular prism is the product of its sides), we can interpret dV = ρ 2 sin φ dρ dφ dθ as the volume of the spherical differential volume element.
When the Class 12 was introduced, it was the largest non-articulated engine in South Africa and possibly in the world on railways of less than 4 feet 8 + 1 ⁄ 2 inches (1,435 millimetres) gauge. The first eight locomotives had cast steel frames for the leading bogies with three-point suspension links, similar to those of the Class 3B .
Instead, the determinant can be evaluated in () operations by forming the LU decomposition = (typically via Gaussian elimination or similar methods), in which case = and the determinants of the triangular matrices and are simply the products of their diagonal entries. (In practical applications of numerical linear algebra, however, explicit ...
Gottfried Wilhelm Leibniz (or Leibnitz; [a] 1 July 1646 [O.S. 21 June] – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many other branches of mathematics, such as binary arithmetic and statistics.