For each of the following statements, indicate whether the statement is true or false. If a triangular matrix has a zero on its main diagonal, then it is necessarily singular. The product of two upper triangular matrices is upper triangular. If a linear system is well-conditioned, then pivoting is unnecessary in Gaussian elimination. Once the LU factorization of a matrix has been computed to solve a linear system, then subsequent linear systems with the same matrix but different right-hand-side vectors can be solved without refactoring the matrix