The authors derive lower and upper bounds for \(\text{tr}(A^{-1})\) and \(\text{det}(A)\) of a symmetric positive definite matrix \(A\), using Gaussian quadrature and related theory. The bounds for \(\text{det}(A)\) are new. The bounds for \(\text{tr}(A^{-1})\) are directly derived instead of the summation of bounds for each diagonal entry of \(A^{-1}\) and are tighter when simple trial vectors are used in Kantorovich’s bounds and are equivalent to Robinson and Wathen’s variational bounds. These bounds are poorer than the probabilistic bounds, but computationally easier and cheaper.