DETERMINANTLAR, USTIDAGI AMALLARNING XOSSALARINI ISBOTLASH
Keywords:
Kalit so‘zlar Determinantlar Nazariyasi, determinantning xossalari, algebraik determinantlar, geometrik talqin, minorlar va kofaktorlar, Laplas yoyilmasi, Gauss eliminasiyasi, LU-faktorizatsiya, Jacobian, Vandermonde determinant., Keywords: Determinant Theory, properties of determinants, algebraic determinants, geometric interpretation, minors and cofactors, Laplace expansion, Gauss elimination, LU-factorization, Jacobian, Vandermonde determinant.Abstract
Annotatsiya: Ushbu kengaytirilgan ilmiy maqolada determinantlar nazariyasining asosiy tushunchalari, tarixiy rivoji, matematik mohiyati, algebraik xossalari, geometrik talqinlari, hisoblashning klassik va zamonaviy usullari, minorlar tizimi, Laplas yoyilmasi, Gauss eliminasiyasi, LU-faktorizatsiya, Jacobian, Vandermonde determinanti, Gram determinanti kabi ko‘plab yo‘nalishlar chuqur yoritilgan. Shuningdek, determinantlarning fizika, mexanika, injiniring, iqtisodiyot, statistika, kompyuter grafikasi, transformatsiya nazariyasi, algoritmlar, kriptografiya va boshqa zamonaviy fanlardagi amaliy qo‘llanilish sohalari ham keng yoritilgan.
Abstract: This extensive scientific article covers in depth the basic concepts of determinant theory, its historical development, mathematical essence, algebraic properties, geometric interpretations, classical and modern methods of calculation, the system of minors, Laplace expansion, Gauss elimination, LU-factorization, Jacobian, Vandermonde determinant, Gram determinant, and many other areas. It also covers the practical applications of determinants in physics, mechanics, engineering, economics, statistics, computer graphics, transformation theory, algorithms, cryptography, and other modern sciences.
