np.linalg.det says nothing about singularity; solve doesn't raise

NumPy 2.5.4 with OpenBLAS. Take a random normal n x n matrix and make its last column an exact linear combination of the others, so it is rank n-1 in exact arithmetic. det was 7.0e15 at n=50, -5.8e173 at n=200 and inf (overflow warning) at n=800. matrix_rank reports n-1 and cond is 2.6e17-7e17. A well-conditioned random matrix (cond 1.3e2 at n=50) has det -2.5e31, so the size of det separates nothing; det also scales as c^n, with det(0.1*I_400) = 0.0 and det(10*I_400) = inf. np.linalg.solve on the singular matrices did not raise LinAlgError. It returned |x| = 1.8e15 (n=50) with relative residual 1.7 for a generic right-hand side. LinAlgError only comes when LU hits an exact zero pivot. To test singularity, use np.linalg.cond or the SVD ratio s_min/s_max against about n*eps, or matrix_rank. For likelihood work, use slogdet (logabsdet 921.03 for 10*I_400 where det overflows). Also, det of [[1,2,3],[4,5,6],[7,8,9]] gave exactly 0.0 on NumPy 1.26.4, 2.1.3 and 2.5.4 here, but that is a lucky LU rounding, and cond for the same matrix is 5e16. Don't use det == 0 as a test.