(27) (10%) Let \( \mathbf{A} \) be a symmetric positive definite matrix with Cholesky decomposition \( \mathbf{A} = \mathbf{R}_{1}^{\text{T}}\mathbf{R}_{1} = \mathbf{R}_{2}^{\text{T}}\mathbf{R}_{2} \). By using the uniqueness of LDU factorization, prove that the upper triangular matrices \( \mathbf{R}_{1} = \mathbf{R}_{2} \).