diff --git a/jax/_src/numpy/linalg.py b/jax/_src/numpy/linalg.py
index cd000862bacc..976c509a7f54 100644
--- a/jax/_src/numpy/linalg.py
+++ b/jax/_src/numpy/linalg.py
@@ -465,6 +465,67 @@ def matrix_rank(
   return reductions.sum(S > rtol, axis=-1)
 
 
+def _slogdet_from_det(det: Array, a: Array) -> tuple[Array, Array]:
+  """Helper to compute slogdet from a pre-computed determinant."""
+  dtype = lax.dtype(a)
+  abs_det = ufuncs.abs(det)
+  is_zero = abs_det == jnp.array(0, dtype=dtype)
+  if jnp.iscomplexobj(a):
+    sign = jnp.where(is_zero, jnp.array(0, dtype=dtype), det / ufuncs.astype(abs_det, det.dtype))
+  else:
+    sign = jnp.where(is_zero, jnp.array(0, dtype=dtype), ufuncs.sign(det))
+  logabsdet = jnp.where(
+      is_zero, jnp.array(-np.inf, dtype=dtype), ufuncs.log(abs_det).astype(dtype))
+  return sign, ufuncs.real(logabsdet)
+
+
+def _slogdet_1x1(a: Array) -> tuple[Array, Array]:
+  """Analytic slogdet for 1x1 matrices. No LU/solve; avoids TRSV on ROCm."""
+  det = a[..., 0, 0]
+  return _slogdet_from_det(det, a)
+
+
+def _slogdet_2x2(a: Array) -> tuple[Array, Array]:
+  """Analytic slogdet for 2x2 matrices. No LU/solve; avoids TRSV on ROCm."""
+  a00, a01 = a[..., 0, 0], a[..., 0, 1]
+  a10, a11 = a[..., 1, 0], a[..., 1, 1]
+  det = (a00 * a11) - (a01 * a10)
+  return _slogdet_from_det(det, a)
+
+
+@custom_jvp
+def _slogdet_small(a: Array) -> tuple[Array, Array]:
+  """slogdet for n in {1, 2} using analytic formulas only (no solve)."""
+  n = a.shape[-1]
+  if n == 1:
+    return _slogdet_1x1(a)
+  elif n == 2:
+    return _slogdet_2x2(a)
+  else:
+    raise ValueError(f"_slogdet_small only supports n in {{1, 2}}, got n={n}")
+
+
+def _slogdet_small_jvp(primals, tangents):
+  x, = primals
+  g, = tangents
+  n = x.shape[-1]
+  sign, ans = _slogdet_small(x)
+  if n == 1:
+    ans_dot = ufuncs.real(g[..., 0, 0] / x[..., 0, 0])
+  else:  # n == 2
+    x00, x01 = x[..., 0, 0], x[..., 0, 1]
+    x10, x11 = x[..., 1, 0], x[..., 1, 1]
+    det = (x00 * x11) - (x01 * x10)
+    ans_dot = ufuncs.real(
+        (g[..., 0, 0] * x11 - g[..., 0, 1] * x10 - g[..., 1, 0] * x01 + g[..., 1, 1] * x00) / det
+    )
+  sign_dot = array_creation.zeros_like(sign)
+  return (sign, ans), (sign_dot, ans_dot)
+
+
+_slogdet_small.defjvp(_slogdet_small_jvp)
+
+
 @custom_jvp
 def _slogdet_lu(a: Array) -> tuple[Array, Array]:
   dtype = lax.dtype(a)
@@ -548,6 +609,9 @@ def slogdet(a: ArrayLike, *, method: str | None = None) -> SlogdetResult:
   if len(a_shape) < 2 or a_shape[-1] != a_shape[-2]:
     raise ValueError(f"Argument to slogdet() must have shape [..., n, n], got {a_shape}")
   if method is None or method == "lu":
+    n = a_shape[-1]
+    if n == 1 or n == 2:
+      return SlogdetResult(*_slogdet_small(a))
     return SlogdetResult(*_slogdet_lu(a))
   elif method == "qr":
     return SlogdetResult(*_slogdet_qr(a))