diff --git a/src/dispatch.jl b/src/dispatch.jl
index 725c7b3f..3ffa4e36 100644
--- a/src/dispatch.jl
+++ b/src/dispatch.jl
@@ -343,199 +343,35 @@ end
 # `LinearAlgebra.mul!` which prevents us from using mutability of the
 # arithmetic. For this reason we intercept the calls and redirect them to `mul`.
 
-# A few are overwritten below but many more need to be redirected to `mul` in
-# `linalg.jl`.
-
-Base.:*(A::_SparseMat{<:AbstractMutable}, x::StridedVector) = mul(A, x)
-
-Base.:*(A::_SparseMat, x::StridedVector{<:AbstractMutable}) = mul(A, x)
-
-function Base.:*(
-    A::_SparseMat{<:AbstractMutable},
-    x::StridedVector{<:AbstractMutable},
-)
-    return mul(A, x)
-end
-
-# These six methods are needed on Julia v1.2 and earlier
-function Base.:*(
-    A::LinearAlgebra.Adjoint{<:AbstractMutable,<:_SparseMat},
-    x::StridedVector,
-)
-    return mul(A, x)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Adjoint{<:Any,<:_SparseMat},
-    x::StridedVector{<:AbstractMutable},
-)
-    return mul(A, x)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Adjoint{<:AbstractMutable,<:_SparseMat},
-    x::StridedVector{<:AbstractMutable},
-)
-    return mul(A, x)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Transpose{<:AbstractMutable,<:_SparseMat},
-    x::StridedVector,
-)
-    return mul(A, x)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Transpose{<:Any,<:_SparseMat},
-    x::StridedVector{<:AbstractMutable},
-)
-    return mul(A, x)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Transpose{<:AbstractMutable,<:_SparseMat},
-    x::StridedVector{<:AbstractMutable},
-)
-    return mul(A, x)
-end
-
-function Base.:*(
-    A::_SparseMat{<:AbstractMutable},
-    B::_SparseMat{<:AbstractMutable},
-)
-    return mul(A, B)
-end
-
-Base.:*(A::_SparseMat{<:Any}, B::_SparseMat{<:AbstractMutable}) = mul(A, B)
-
-Base.:*(A::_SparseMat{<:AbstractMutable}, B::_SparseMat{<:Any}) = mul(A, B)
-
-function Base.:*(
-    A::_SparseMat{<:AbstractMutable},
-    B::LinearAlgebra.Adjoint{<:AbstractMutable,<:_SparseMat},
-)
-    return mul(A, B)
-end
-
-function Base.:*(
-    A::_SparseMat{<:Any},
-    B::LinearAlgebra.Adjoint{<:AbstractMutable,<:_SparseMat},
-)
-    return mul(A, B)
-end
-
-function Base.:*(
-    A::_SparseMat{<:AbstractMutable},
-    B::LinearAlgebra.Adjoint{<:Any,<:_SparseMat},
-)
-    return mul(A, B)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Adjoint{<:AbstractMutable,<:_SparseMat},
-    B::_SparseMat{<:AbstractMutable},
-)
-    return mul(A, B)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Adjoint{<:Any,<:_SparseMat},
-    B::_SparseMat{<:AbstractMutable},
-)
-    return mul(A, B)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Adjoint{<:AbstractMutable,<:_SparseMat},
-    B::_SparseMat{<:Any},
-)
-    return mul(A, B)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Transpose{<:AbstractMutable,<:_SparseMat},
-    B::_SparseMat{<:AbstractMutable},
-)
-    return mul(A, B)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Transpose{<:Any,<:_SparseMat},
-    B::_SparseMat{<:AbstractMutable},
-)
-    return mul(A, B)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Transpose{<:AbstractMutable,<:_SparseMat},
-    B::_SparseMat{<:Any},
-)
-    return mul(A, B)
-end
-
-function Base.:*(
-    A::StridedMatrix{<:AbstractMutable},
-    B::_SparseMat{<:AbstractMutable},
-)
-    return mul(A, B)
-end
-
-Base.:*(A::StridedMatrix{<:Any}, B::_SparseMat{<:AbstractMutable}) = mul(A, B)
-
-Base.:*(A::StridedMatrix{<:AbstractMutable}, B::_SparseMat{<:Any}) = mul(A, B)
-
-function Base.:*(
-    A::_SparseMat{<:AbstractMutable},
-    B::StridedMatrix{<:AbstractMutable},
-)
-    return mul(A, B)
-end
-
-Base.:*(A::_SparseMat{<:Any}, B::StridedMatrix{<:AbstractMutable}) = mul(A, B)
-
-Base.:*(A::_SparseMat{<:AbstractMutable}, B::StridedMatrix{<:Any}) = mul(A, B)
-
-function Base.:*(
-    A::LinearAlgebra.Adjoint{<:AbstractMutable,<:_SparseMat},
-    B::StridedMatrix{<:AbstractMutable},
-)
-    return mul(A, B)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Adjoint{<:Any,<:_SparseMat},
-    B::StridedMatrix{<:AbstractMutable},
-)
-    return mul(A, B)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Adjoint{<:AbstractMutable,<:_SparseMat},
-    B::StridedMatrix{<:Any},
-)
-    return mul(A, B)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Transpose{<:AbstractMutable,<:_SparseMat},
-    B::StridedMatrix{<:AbstractMutable},
-)
-    return mul(A, B)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Transpose{<:Any,<:_SparseMat},
-    B::StridedMatrix{<:AbstractMutable},
-)
-    return mul(A, B)
-end
-
-function Base.:*(
-    A::LinearAlgebra.Transpose{<:AbstractMutable,<:_SparseMat},
-    B::StridedMatrix{<:Any},
-)
-    return mul(A, B)
+const _LinearAlgebraWrappers = (
+    LinearAlgebra.Adjoint,
+    LinearAlgebra.Transpose,
+    # TODO(odow): we could expand these overloads to other LinearAlgebra types.
+    # LinearAlgebra.Symmetric,
+    # LinearAlgebra.Hermitian,
+    # LinearAlgebra.Diagonal,
+    # LinearAlgebra.LowerTriangular,
+    # LinearAlgebra.UpperTriangular,
+    # LinearAlgebra.UnitLowerTriangular,
+    # LinearAlgebra.UnitUpperTriangular,
+)
+
+const _MatrixLike = vcat(
+    Any[T -> LA{<:T,<:_SparseMat} for LA in _LinearAlgebraWrappers],
+    Any[T->_SparseMat{<:T}, T->StridedMatrix{<:T}],
+)
+
+for f_A in _MatrixLike, f_B in vcat(_MatrixLike, T -> StridedVector{<:T})
+    A, mut_A = f_A(Any), f_A(AbstractMutable)
+    B, mut_B = f_B(Any), f_B(AbstractMutable)
+    if A <: StridedMatrix && B <: StridedMatrix
+        continue
+    end
+    @eval begin
+        Base.:*(a::$(mut_A), b::$(B)) = mul(a, b)
+        Base.:*(a::$(A), b::$(mut_B)) = mul(a, b)
+        Base.:*(a::$(mut_A), b::$(mut_B)) = mul(a, b)
+    end
 end
 
 const StridedMaybeAdjOrTransMat{T} = Union{