diff --git a/lineax/_operator.py b/lineax/_operator.py
index b430a2f..fc05c23 100644
--- a/lineax/_operator.py
+++ b/lineax/_operator.py
@@ -1148,12 +1148,20 @@ def mv(self, vector):
         return self.operator1.mv(self.operator2.mv(vector))
 
     def as_matrix(self):
-        return jnp.matmul(
-            self.operator1.as_matrix(),
-            self.operator2.as_matrix(),
-            precision=lax.Precision.HIGHEST,  # pyright: ignore
+        if isinstance(self.operator1, IdentityLinearOperator):
+            return self.operator2.as_matrix()
+        if isinstance(self.operator2, IdentityLinearOperator):
+            return self.operator1.as_matrix()
+        _, unravel = eqx.filter_eval_shape(
+            jfu.ravel_pytree, self.operator1.in_structure()
         )
 
+        def mv_flat(v):
+            out = self.operator1.mv(unravel(v))
+            return jfu.ravel_pytree(out)[0]
+
+        return jax.vmap(mv_flat, in_axes=1, out_axes=1)(self.operator2.as_matrix())
+
     def transpose(self):
         return self.operator2.transpose() @ self.operator1.transpose()
 
@@ -1939,7 +1947,13 @@ def _(operator):
 
 @materialise.register(ComposedLinearOperator)
 def _(operator):
-    return materialise(operator.operator1) @ materialise(operator.operator2)
+    op1 = materialise(operator.operator1)
+    op2 = materialise(operator.operator2)
+    if isinstance(op1, IdentityLinearOperator):
+        return op2
+    if isinstance(op2, IdentityLinearOperator):
+        return op1
+    return op1 @ op2
 
 
 @diagonal.register(ComposedLinearOperator)