Final answer:
To answer this question, we need to perform various operations on the given system of linear equations. First, we arrange the system in matrix form and find its inverse using Gauss-Jordan elimination and the cofactor method. Then, we evaluate the determinant using the cofactor method and find the inverse using the determinant and the adjoint of the matrix. We also factorize the matrix and solve the system using both the factorization method and Cramer's Rule.
Step-by-step explanation:
To write the given system of linear equations in the form A, we arrange the coefficients of x, y, and z into a matrix:
A = [1 -3 0; 0 -5 3; 1 -10 2].
To find the inverse of A using Gauss-Jordan elimination, we perform row operations on the augmented matrix of A to transform it into reduced row echelon form. The resulting matrix will be the inverse of A.
To evaluate the determinant of A using the cofactor method, we use the formula det(A) = a11C11 + a12C12 + a13C13, where aij represents the element of A in the i-th row and j-th column, and Cij represents the cofactor of that element.
To find the inverse of A using the determinant and the adjoint of A, we use the formula A^-1 = (1/det(A)) * adj(A), where adj(A) represents the adjoint of A.
To factorize A, we find a matrix B such that A = LU, where L is a lower triangular matrix and U is an upper triangular matrix.
To solve the system of equations using the factorization method, we substitute A = LU into the original system of equations and solve for the unknowns.
To solve the system of equations using Cramer's Rule, we find the determinants of the coefficient matrix and each augmented matrix formed by replacing one column of the coefficient matrix with the column of constants. Then, we divide each resultant determinant by the determinant of the coefficient matrix to obtain the values of the unknowns.
By comparing the solutions obtained through the factorization method and Cramer's Rule, we can verify if they coincide.