diff --git a/src/Test/array.jl b/src/Test/array.jl
index f3875b4..7ac60f6 100644
--- a/src/Test/array.jl
+++ b/src/Test/array.jl
@@ -8,21 +8,35 @@ function matrix_vector_division_test(x)
 end
 
 function _xAx_test(x::AbstractVector, A::AbstractMatrix)
-    @test_rewrite(x' * A)
-    # Complex expression
-    @test_rewrite(x' * ones(Int, size(A)...))
-    @test_rewrite(x' * A * x)
-    # Complex expression
-    @test_rewrite(x' * ones(Int, size(A)...) * x)
-    @test_rewrite reshape(x, (1, length(x))) * A * x .- 1
-    @test_rewrite x' * A * x .- 1
-    @test_rewrite x' * A * x - 1
+    for t in [transpose, adjoint]
+        @test_rewrite(t(x) * A)
+        # Complex expression
+        @test_rewrite(t(x) * ones(Int, size(A)...))
+        @test_rewrite(t(x) * A * x)
+        # Complex expression
+        @test_rewrite(t(x) * ones(Int, size(A)...) * x)
+        @test_rewrite reshape(x, (1, length(x))) * A * x .- 1
+        @test_rewrite t(x) * A * x .- 1
+        @test_rewrite t(x) * A * x - 1
+        @test_rewrite t(x) * x + t(x) * A * x
+        @test_rewrite t(x) * x - t(x) * A * x
+        @test MA.promote_operation(*, typeof(t(x)), typeof(A), typeof(x)) == typeof(t(x) * A * x)
+        @test MA.promote_operation(*, typeof(t(x)), typeof(x)) == typeof(t(x) * x)
+        @test_rewrite t(x) * x + 2 * t(x) * A * x
+        @test_rewrite t(x) * x - 2 * t(x) * A * x
+        @test_rewrite t(x) * A * x + 2 * t(x) * x
+        @test_rewrite t(x) * A * x - 2 * t(x) * x
+        @test MA.promote_operation(*, Int, typeof(t(x)), typeof(A), typeof(x)) == typeof(2 * t(x) * A * x)
+        @test MA.promote_operation(*, Int, typeof(t(x)), typeof(x)) == typeof(2 * t(x) * x)
+    end
 end
 function _xABx_test(x::AbstractVector, A::AbstractMatrix, B::AbstractMatrix)
-    @test_rewrite (x'A)' + 2B * x
-    @test_rewrite (x'A)' + 2B * x .- 1
-    @test_rewrite (x'A)' + 2B * x .- [length(x):-1:1;]
-    @test_rewrite (x'A)' + 2B * x - [length(x):-1:1;]
+    for t in [transpose, adjoint]
+        @test_rewrite t(t(x) * A) + 2B * x
+        @test_rewrite t(t(x) * A) + 2B * x .- 1
+        @test_rewrite t(t(x) * A) + 2B * x .- [length(x):-1:1;]
+        @test_rewrite t(t(x) * A) + 2B * x - [length(x):-1:1;]
+    end
 end
 
 function _matrix_vector_test(x::AbstractVector, A::AbstractMatrix)
diff --git a/src/linear_algebra.jl b/src/linear_algebra.jl
index f7f8250..f3a54e6 100644
--- a/src/linear_algebra.jl
+++ b/src/linear_algebra.jl
@@ -233,10 +233,17 @@ end
 const TransposeOrAdjoint{T, MT} = Union{LinearAlgebra.Transpose{T, MT}, LinearAlgebra.Adjoint{T, MT}}
 _mirror_transpose_or_adjoint(x, ::LinearAlgebra.Transpose) = LinearAlgebra.transpose(x)
 _mirror_transpose_or_adjoint(x, ::LinearAlgebra.Adjoint) = LinearAlgebra.adjoint(x)
+_mirror_transpose_or_adjoint(A::Type{<:AbstractArray{T}}, ::Type{<:LinearAlgebra.Transpose}) where {T} = LinearAlgebra.Transpose{T, A}
+_mirror_transpose_or_adjoint(A::Type{<:AbstractArray{T}}, ::Type{<:LinearAlgebra.Adjoint}) where {T} = LinearAlgebra.Adjoint{T, A}
+similar_array_type(TA::Type{<:TransposeOrAdjoint{T, A}}, ::Type{S}) where {S, T, A} = _mirror_transpose_or_adjoint(similar_array_type(A, S), TA)
 # dot product
 function promote_array_mul(::Type{<:TransposeOrAdjoint{S, <:AbstractVector}}, ::Type{<:AbstractVector{T}}) where {S, T}
     return promote_sum_mul(S, T)
 end
+function promote_array_mul(A::Type{<:TransposeOrAdjoint{S, V}}, M::Type{<:AbstractMatrix{T}}) where {S, T, V <: AbstractVector}
+    B = promote_array_mul(_mirror_transpose_or_adjoint(M, A), V)
+    return _mirror_transpose_or_adjoint(B, A)
+end
 function operate(::typeof(*), x::LinearAlgebra.Adjoint{<:Any, <:AbstractVector}, y::AbstractVector)
     return operate(LinearAlgebra.dot, parent(x), y)
 end
diff --git a/test/dummy.jl b/test/dummy.jl
index be23c03..2bd90a0 100644
--- a/test/dummy.jl
+++ b/test/dummy.jl
@@ -30,8 +30,8 @@ MA.scaling(x::DummyBigInt) = x
 
 MA.mutable_operate_to!(x::DummyBigInt, op::Function, args::Union{MA.Scaling, DummyBigInt}...) = DummyBigInt(MA.mutable_operate_to!(x.data, op, _data.(args)...))
 # Called for instance if `args` is `(v', v)` for a vector `v`.
-MA.mutable_operate_to!(output::DummyBigInt, op::typeof(MA.add_mul), x::Union{MA.Scaling, DummyBigInt}, y::Union{MA.Scaling, DummyBigInt}, z::Union{MA.Scaling, DummyBigInt}, args::Union{MA.Scaling, DummyBigInt}...) = MA.mutable_operate_to!(output, +, x, *(y, z, args...))
-MA.mutable_operate_to!(output::DummyBigInt, op::typeof(MA.add_mul), x, y, z, args...) = MA.mutable_operate_to!(output, +, x, *(y, z, args...))
+MA.mutable_operate_to!(output::DummyBigInt, op::MA.AddSubMul, x::Union{MA.Scaling, DummyBigInt}, y::Union{MA.Scaling, DummyBigInt}, z::Union{MA.Scaling, DummyBigInt}, args::Union{MA.Scaling, DummyBigInt}...) = MA.mutable_operate_to!(output, MA.add_sub_op(op), x, *(y, z, args...))
+MA.mutable_operate_to!(output::DummyBigInt, op::MA.AddSubMul, x, y, z, args...) = MA.mutable_operate_to!(output, MA.add_sub_op(op), x, *(y, z, args...))
 function MA.mutable_operate!(op::Function, x::DummyBigInt, args::Vararg{Any, N}) where N
     MA.mutable_operate_to!(x, op, x, args...)
 end