題組內容

(a)(5%) The nonsingular matrix A can be reduced to row echelon form with a finite number of elementary row operations. Thus \( U = E_k E_{k-1} \cdots E_1 A \), where \( U \) denotes the upper triangular matrix, and \( E_n \) represents the elementary matrix, \( n = 1, \dots, k \). Then \( \det(U) = \det(E_k)\det(E_{k-1}) \cdots \det(E_1)\det(A) \). Use the above two equations, first find \( U \) and then compute \( \det(A) \) for the above linear system.