diff --git a/src/MutableArithmetics.jl b/src/MutableArithmetics.jl
index 42b5871..bcf4cfe 100644
--- a/src/MutableArithmetics.jl
+++ b/src/MutableArithmetics.jl
@@ -107,6 +107,9 @@ end
 function isequal_canonical(x::LinearAlgebra.Tridiagonal, y::LinearAlgebra.Tridiagonal)
     return isequal_canonical(x.dl, y.dl) && isequal_canonical(x.d, y.d) && isequal_canonical(x.du, y.du)
 end
+function isequal_canonical(x::SparseMat, y::SparseMat)
+    return x.m == y.m && x.n == y.n && isequal_canonical(x.colptr, y.colptr) && isequal_canonical(x.rowval, y.rowval) && isequal_canonical(x.nzval, y.nzval)
+end
 
 include("rewrite.jl")
 include("dispatch.jl")
diff --git a/src/Test/sparse.jl b/src/Test/sparse.jl
index f1f7aa4..5488679 100644
--- a/src/Test/sparse.jl
+++ b/src/Test/sparse.jl
@@ -15,6 +15,21 @@ function sparse_linear_test(X11, X23, Xd)
           0    0 4X23
           0    0    0]
 
+    function _test_broadcast(A, B)
+        @test_rewrite(A .* B)
+        @test_rewrite(B .* A)
+        @test_rewrite(A .+ B)
+        @test_rewrite(B .+ A)
+        @test_rewrite(A .- B)
+        @test_rewrite(B .- A)
+    end
+
+    _test_broadcast(Xd, X)
+    _test_broadcast(Xd, Y)
+    _test_broadcast(Yd, X)
+    _test_broadcast(Yd, Y)
+    _test_broadcast(X, Y)
+
     add_test(Xd, Yd)
     add_test(Xd, Y)
     add_test(Xd, Xd)
diff --git a/src/broadcast.jl b/src/broadcast.jl
index cc2a62a..8fe978b 100644
--- a/src/broadcast.jl
+++ b/src/broadcast.jl
@@ -1,7 +1,7 @@
-function broadcasted_type(::Broadcast.DefaultArrayStyle{N}, ::Type{Eltype}) where {N, Eltype}
+function broadcasted_type(::Broadcast.DefaultArrayStyle{N}, ::Base.HasShape{N}, ::Type{Eltype}) where {N, Eltype}
     return Array{Eltype, N}
 end
-function broadcasted_type(::Broadcast.DefaultArrayStyle{N}, ::Type{Bool}) where N
+function broadcasted_type(::Broadcast.DefaultArrayStyle{N}, ::Base.HasShape{N}, ::Type{Bool}) where N
     return BitArray{N}
 end
 
@@ -11,12 +11,18 @@ combine_styles(c::Type) = Broadcast.result_style(Broadcast.BroadcastStyle(c))
 combine_styles(c1::Type, c2::Type) = Broadcast.result_style(combine_styles(c1), combine_styles(c2))
 @inline combine_styles(c1::Type, c2::Type, cs::Vararg{Type, N}) where N = Broadcast.result_style(combine_styles(c1), combine_styles(c2, cs...))
 
+combine_shapes(s) = s
+combine_2_shapes(s1::Base.HasShape{N}, s2::Base.HasShape{M}) where {N, M} = Base.HasShape{max(N, M)}()
+combine_shapes(s1, s2, args::Vararg{Any, N}) where {N} = combine_shapes(combine_2_shapes(s1, s2), args...)
+_shape(T) = Base.HasShape{ndims(T)}()
+combine_sizes(args::Vararg{Any, N}) where {N} = combine_shapes(_shape.(args)...)
+
 function promote_broadcast(op::Function, args::Vararg{Any, N}) where N
     # FIXME we could use `promote_operation` instead as
     # `combine_eltypes` uses `return_type` hence it may return a non-concrete type
     # and we do not handle that case.
     T = Base.Broadcast.combine_eltypes(op, args)
-    return broadcasted_type(combine_styles(args...), T)
+    return broadcasted_type(combine_styles(args...), combine_sizes(args...), T)
 end
 
 """
diff --git a/src/sparse_arrays.jl b/src/sparse_arrays.jl
index 39b033e..9a291e5 100644
--- a/src/sparse_arrays.jl
+++ b/src/sparse_arrays.jl
@@ -220,3 +220,12 @@ function mutable_operate!(::typeof(add_mul), ret::SparseMat{T},
                           α::Vararg{Union{T, Scaling}, N}) where {T, N}
     mutable_operate!(add_mul, ret, copy(A), B, α...)
 end
+
+# This `BroadcastStyle` is used when there is a mix of sparse arrays and dense arrays.
+# The result is a sparse array.
+function broadcasted_type(::SparseArrays.HigherOrderFns.PromoteToSparse, ::Base.HasShape{1}, ::Type{Eltype}) where Eltype
+    return SparseArrays.SparseVector{Eltype, Int}
+end
+function broadcasted_type(::SparseArrays.HigherOrderFns.PromoteToSparse, ::Base.HasShape{2}, ::Type{Eltype}) where Eltype
+    return SparseMat{Eltype, Int}
+end