試題詳解

1. When using the Gaussian elimination to solve a linear equation Ax = b, elementary row operations (or multiplication by elementary matrices) are applied to the augmented matrix [A b]. Actually, there are many places in linear algebra where such a technique plays its role； e.g., the null space of A can be determined by setting b = 0. Which of the following cannot be determined by applying such a technique?
(A) the range of A
(B) the inverse of A (if it is nonsingular)
(C) the QR factorization of A
(D) the LU factorization of A (if it is square)
(E) the determinant of A (if it is square).