diff --git a/src/matmul.jl b/src/matmul.jl
index 58bad14..64187c7 100644
--- a/src/matmul.jl
+++ b/src/matmul.jl
@@ -53,15 +53,19 @@ for T in (Complex{Float32}, Complex{Float64}, Float32, Float64),
             Base.:(*)(A::$AT, B::StridedMatrix{$T}) = mul!(Matrix{$T}(undef, size(A, 1), size(B, 2)), A, B)
         end
 
+        # define 4-arg ldiv!(C, A, B, a) (C := alpha*inv(A)*B) that is not present in standard LinearAlgrebra,
+        # redefine 3-arg ldiv!(C, A, B) using 4-arg ldiv!(C, A, B, 1)
+        # here A is LowerTri/UpperTri etc of a SparseMatrixCSC or adjoint/transpose of it (Symmetric not supported)
         if w != :Symmetric
             @eval begin
-                LinearAlgebra.ldiv!(α::Number, A::$AT, B::StridedVector{$T}, C::StridedVector{$T}) =
+                LinearAlgebra.ldiv!(C::StridedVector{$T}, A::$AT, B::StridedVector{$T}, α::Number) =
                     cscsv!($tchar, $T(α), describe_and_unwrap(A)..., B, C)
-                LinearAlgebra.ldiv!(α::Number, A::$AT, B::StridedMatrix{$T}, C::StridedMatrix{$T}) =
+                LinearAlgebra.ldiv!(C::StridedMatrix{$T}, A::$AT, B::StridedMatrix{$T}, α::Number) =
                     cscsm!($tchar, $T(α), describe_and_unwrap(A)..., B, C)
 
-                LinearAlgebra.ldiv!(C::BT, A::$AT, B::BT) where BT <: $BT = ldiv!(one($T), A, B, C)
+                LinearAlgebra.ldiv!(C::BT, A::$AT, B::BT) where BT <: $BT = ldiv!(C, A, B, one($T))
 
+                # base A\B converts A to dense, so we redefine it to use MKLSparse-enabled ldiv!
                 Base.:(\)(A::$AT, B::StridedVector{$T}) = ldiv!(Vector{$T}(undef, size(A, 1)), A, B)
                 Base.:(\)(A::$AT, B::StridedMatrix{$T}) = ldiv!(Matrix{$T}(undef, size(A, 1), size(B, 2)), A, B)
             end
diff --git a/test/test_BLAS.jl b/test/test_BLAS.jl
index 55066ce..c36daae 100644
--- a/test/test_BLAS.jl
+++ b/test/test_BLAS.jl
@@ -144,7 +144,7 @@ end
         spAclass = Aclass(spA)
         α = rand()
 
-        @test @blas(ldiv!(α, Aclass(spA), B, similar(B))) ≈ α * (Aclass(A) \ B)
+        @test @blas(ldiv!(similar(B), Aclass(spA), B, α)) ≈ α * (Aclass(A) \ B)
         @test @blas(ldiv!(similar(B), Aclass(spA), B)) ≈ Aclass(A) \ B
         @test @blas(Aclass(spA) \ B) ≈ Aclass(A) \ B
     end