Cálculo Matricial

Permutação σ ∈ S 3 sgn(σ) a 1σ(1) a 2σ(2) · · · a nσ(n)(123) +1 a 11 a 22 a 33(231) +1 a 12 a 23 a 31(312) +1 a 13 a 21 a 32(132) −1 a 11 a 23 a 32(213) −1 a 12 a 21 a 33(321) −1 a 11 a 22 a 31Obtemos, assim,|A| = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32−a 11 a 23 a 32 − a 12 a 21 a 33 − a 11 a 22 a 31Para fácil memorização, pode-se recorrer ao esquema apresentado de seguida.00000000000000000000011111111111111111111100000000000000000000011111111111111111111100000000000000000000011111111111111111111100000000000000000000011111111111111111111100000000000000000000000111111111111111111111110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100000000000000000000011111111111111111111100000000000000000000011111111111111111111100000000000000000000000111111111111111111111110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100000000000000000000011111111111111111111100 1100 1100000000000000000000011111111111111111111100000000000000000000000111111111111111111111110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100000000000000000000011111111111111111111100 1100 1100000000000000000000011111111111111111111100000000000000000000000111111111111111111111110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100000000000000000000011111111111111111111100000000000000000000011111111111111111111100000000000000000000000111111111111111111111110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100000000000000000000011111111111111111111100000000000000000000011111111111111111111100000000000000000000000111111111111111111111110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100000000000000000000011111111111111111111100000000000000000000011111111111111111111100000000000000000000000111111111111111111111110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100000000000000000000011111111111111111111100000000000000000000011111111111111111111100000000000000000000000111111111111111111111110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100 11 00000000000000000000011111111111111111111100 1100000000000000000000011111111111111111111100000000000000000000000111111111111111111111110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100 11 00000000000000000000011111111111111111111100 11000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100000000000000000000011111111111111111111100000000000000000000011111111111111111111100000000000000000000000111111111111111111111110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100000000000000000000011111111111111111111100000000000000000000011111111111111111111100000000000000000000000111111111111111111111110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100000000000000000000011111111111111111111100000000000000000000011111111111111111111100000000000000000000000111111111111111111111110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100000000000000000000011111111111111111111100000000000000000000011111111111111111111100000000000000000000000111111111111111111111110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100000000000000000000011111111111111111111100 11 000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100 110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100000000000000000000011111111111111111111100 11 000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100 11000000000000000000000111111111111111111111000000000000000000000111111111111111111111000000000000000000000001111111111111111111111100000000000000000000011111111111111111111100000000000000000000011111111111111111111100000000000000000000000111111111111111111111110000000000000000000001111111111111111111110000000000000000000001111111111111111111110000000000000000000000011111111111111111111111000000000000000000000001111111111111111111111100000000000000000000000111111111111111111111110000000000000000000000011111111111111111111111Figura 2: Esquema do cálculo do determinante de matrizes de ordem 3, ou a Regra de Sarrus4.2 PropriedadesSão consequência da definição os resultados que de seguida apresentamos, dos quais omitimosa demonstração.Teorema 4.2. Seja A uma matriz quadrada.1. Se A tem uma linha ou uma coluna nula então |A| = 0.2. |A| = |A T |.3. Se A é triangular (inferior ou superior) então |A| = ∏4. |P ij | = −1, |D k (a)| = a, |E ij (a)| = 1, com i ≠ j.i=1,...,n(A) ii .Daqui segue que |I n | = 1. Segue também que dada uma matriz tringular (inferior ou superior)que esta é invertível se e só se tiver determinante não nulo. Mais adiante, apresentaremosum resultado que generaliza esta equivalência para matrizes quadradas não necessariamentetriangulares.Teorema 4.3. Dada uma matriz A quadrada, a ∈ K,36