diff --git a/tests/unit/factored_matrix/test_multiply_by_scalar.py b/tests/unit/factored_matrix/test_multiply_by_scalar.py
new file mode 100644
index 000000000..85d0bfbe7
--- /dev/null
+++ b/tests/unit/factored_matrix/test_multiply_by_scalar.py
@@ -0,0 +1,59 @@
+import random
+
+import pytest
+import torch
+from torch.testing import assert_close
+
+from transformer_lens import FactoredMatrix
+
+
+# This test function is parametrized with different types of scalars, including non-scalar tensors and arrays, to check that the correct errors are raised.
+# Considers cases with and without leading dimensions as well as left and right multiplication.
+@pytest.mark.parametrize(
+    "scalar, error_expected",
+    [
+        # Test cases with different types of scalar values.
+        (torch.rand(1), None),  # 1-element Tensor. No error expected.
+        (random.random(), None),  # float. No error expected.
+        (random.randint(-100, 100), None),  # int. No error expected.
+        # Test cases with non-scalar values that are expected to raise errors.
+        (
+            torch.rand(2, 2),
+            AssertionError,
+        ),  # Non-scalar Tensor. AssertionError expected.
+        (torch.rand(2), AssertionError),  # Non-scalar Tensor. AssertionError expected.
+    ],
+)
+@pytest.mark.parametrize("leading_dim", [False, True])
+@pytest.mark.parametrize("multiply_from_left", [False, True])
+def test_multiply(scalar, leading_dim, multiply_from_left, error_expected):
+    # Prepare a FactoredMatrix, with or without leading dimensions
+    if leading_dim:
+        a = torch.rand(6, 2, 3)
+        b = torch.rand(6, 3, 4)
+    else:
+        a = torch.rand(2, 3)
+        b = torch.rand(3, 4)
+
+    fm = FactoredMatrix(a, b)
+
+    if error_expected:
+        # If an error is expected, check that the correct exception is raised.
+        with pytest.raises(error_expected):
+            if multiply_from_left:
+                _ = fm * scalar
+            else:
+                _ = scalar * fm
+    else:
+        # If no error is expected, check that the multiplication results in the correct value.
+        # Use FactoredMatrix.AB to calculate the product of the two factor matrices before comparing with the expected value.
+        if multiply_from_left:
+            assert_close((fm * scalar).AB, (a @ b) * scalar)
+        else:
+            assert_close((scalar * fm).AB, scalar * (a @ b))
+        # This next test is implementation dependant and can be broken and removed at any time!
+        # It checks that the multiplication is performed on the A factor matrix.
+        if multiply_from_left:
+            assert_close((fm * scalar).A, a * scalar)
+        else:
+            assert_close((scalar * fm).A, scalar * a)
diff --git a/transformer_lens/FactoredMatrix.py b/transformer_lens/FactoredMatrix.py
index cd357b779..3f589ebfd 100644
--- a/transformer_lens/FactoredMatrix.py
+++ b/transformer_lens/FactoredMatrix.py
@@ -82,6 +82,22 @@ def __rmatmul__(
         elif isinstance(other, FactoredMatrix):
             return other.A @ (other.B @ self)
 
+    def __mul__(self, scalar: Union[int, float, torch.Tensor]) -> FactoredMatrix:
+        """
+        Left scalar multiplication. Scalar multiplication distributes over matrix multiplication, so we can just multiply one of the factor matrices by the scalar.
+        """
+        if isinstance(scalar, torch.Tensor):
+            assert (
+                scalar.numel() == 1
+            ), f"Tensor must be a scalar for use with * but was of shape {scalar.shape}. For matrix multiplication, use @ instead."
+        return FactoredMatrix(self.A * scalar, self.B)
+
+    def __rmul__(self, scalar: Union[int, float, torch.Tensor]) -> FactoredMatrix:
+        """
+        Right scalar multiplication. For scalar multiplication from the right, we can reuse the __mul__ method.
+        """
+        return self * scalar
+
     @property
     @typeguard_ignore
     def AB(self) -> Float[torch.Tensor, "*leading_dims ldim rdim"]: