diff --git a/lib/mps/MPS.jl b/lib/mps/MPS.jl
index 04c08f87..10427d1a 100644
--- a/lib/mps/MPS.jl
+++ b/lib/mps/MPS.jl
@@ -24,4 +24,7 @@ include("decomposition.jl")
 # matrix copy
 include("copy.jl")
 
+# solver
+include("solve.jl")
+
 end
diff --git a/lib/mps/linalg.jl b/lib/mps/linalg.jl
index 66944356..e8a952a7 100644
--- a/lib/mps/linalg.jl
+++ b/lib/mps/linalg.jl
@@ -192,3 +192,108 @@ function LinearAlgebra.transpose!(B::MtlMatrix{T}, A::MtlMatrix{T}) where {T}
 
     return B
 end
+
+
+function LinearAlgebra.ldiv!(A::LU{T, <:MtlMatrix{T}}, B::MtlVecOrMat{T}) where {T}
+    # TODO
+end
+
+function LinearAlgebra.ldiv!(A::UpperTriangular{T, <:MtlMatrix{T}}, B::MtlVecOrMat{T}) where {T}
+    M,N = size(B)
+    dev = current_device()
+    queue = global_queue(dev)
+    cmdbuf = MTLCommandBuffer(queue)
+    enqueue!(cmdbuf)
+
+    X = MtlMatrix{T}(undef, size(B))
+
+    mps_a = MPSMatrix(A)
+    mps_b = MPSMatrix(B) # TODO reshape to matrix if B is a vector
+    mps_x = MPSMatrix(X)
+
+    solve_kernel = MPSMatrixSolveTriangular(dev, false, false, false, false, M, N, 1.0) # TODO: likely N, M is the correct order
+    encode!(cmdbuf, solve_kernel, mps_a, mps_b, mps_x)
+    commit!(cmdbuf)
+
+    return X
+end
+
+function LinearAlgebra.ldiv!(A::UnitUpperTriangular{T, <:MtlMatrix{T}}, B::MtlVecOrMat{T}) where {T}
+    M,N = size(B)
+    dev = current_device()
+    queue = global_queue(dev)
+    cmdbuf = MTLCommandBuffer(queue)
+    enqueue!(cmdbuf)
+
+    Bh = reshape(B, )
+    X = MtlMatrix{T}(undef, size(B))
+
+    mps_a = MPSMatrix(A)
+    mps_b = MPSMatrix(Bh) # TODO reshape to matrix if B is a vector
+    mps_x = MPSMatrix(X)
+
+    solve_kernel = MPSMatrixSolveTriangular(dev, false, false, false, true, M, N, 1.0)
+    encode!(cmdbuf, solve_kernel, mps_a, mps_b, mps_x)
+    commit!(cmdbuf)
+
+    return X
+end
+
+function LinearAlgebra.ldiv!(A::LowerTriangular{T, <:MtlMatrix{T}}, B::MtlVecOrMat{T}) where {T}
+    M,N = size(B)
+    dev = current_device()
+    queue = global_queue(dev)
+    cmdbuf = MTLCommandBuffer(queue)
+    enqueue!(cmdbuf)
+
+    X = MtlMatrix{T}(undef, size(B))
+
+    mps_a = MPSMatrix(A)
+    mps_b = MPSMatrix(B) # TODO reshape to matrix if B is a vector
+    mps_x = MPSMatrix(X)
+
+    solve_kernel = MPSMatrixSolveTriangular(dev, false, true, false, false, M, N, 1.0)
+    encode!(cmdbuf, solve_kernel, mps_a, mps_b, mps_x)
+    commit!(cmdbuf)
+
+    return X
+end
+
+function LinearAlgebra.ldiv!(A::UnitLowerTriangular{T, <:MtlMatrix{T}}, B::MtlVecOrMat{T}) where {T}
+    M,N = size(B)
+    dev = current_device()
+    queue = global_queue(dev)
+    cmdbuf = MTLCommandBuffer(queue)
+    enqueue!(cmdbuf)
+
+    X = MtlMatrix{T}(undef, size(B))
+
+    mps_a = MPSMatrix(A)
+    mps_b = MPSMatrix(B) # TODO reshape to matrix if B is a vector
+    mps_x = MPSMatrix(X)
+
+    solve_kernel = MPSMatrixSolveTriangular(dev, false, true, false, true, M, N, 1.0)
+    encode!(cmdbuf, solve_kernel, mps_a, mps_b, mps_x)
+    commit!(cmdbuf)
+
+    return X
+end
+
+# function (\)(A::AbstractMatrix, B::AbstractVecOrMat)
+#     require_one_based_indexing(A, B)
+#     m, n = size(A)
+#     if m == n
+#         if istril(A)
+#             if istriu(A)
+#                 return Diagonal(A) \ B
+#             else
+#                 return LowerTriangular(A) \ B
+#             end
+#         end
+#         if istriu(A)
+#             return UpperTriangular(A) \ B
+#         end
+#         return lu(A) \ B
+#     end
+#     return qr(A, ColumnNorm()) \ B
+# end
\ No newline at end of file
diff --git a/lib/mps/solve.jl b/lib/mps/solve.jl
new file mode 100644
index 00000000..c5978e26
--- /dev/null
+++ b/lib/mps/solve.jl
@@ -0,0 +1,72 @@
+export MPSMatrixSolveLU
+
+@objcwrapper immutable=false MPSMatrixSolveLU <: MPSMatrixBinaryKernel
+
+function MPSMatrixSolveLU(device, transpose, order, numberOfRightHandSides)
+    kernel = @objc [MPSMatrixSolveLU alloc]::id{MPSMatrixSolveLU}
+    obj = MPSMatrixSolveLU(kernel)
+    finalizer(release, obj)
+    @objc [obj::id{MPSMatrixSolveLU} initWithDevice:device::id{MTLDevice}
+                                     transpose:transpose::Bool
+                                     order:order::NSUInteger
+                                     numberOfRightHandSides:numberOfRightHandSides::NSUInteger]::id{MPSMatrixSolveLU}
+    return obj
+end
+
+function encode!(cmdbuf::MTLCommandBuffer, kernel::MPSMatrixSolveLU, sourceMatrix, rightHandSideMatrix, pivotIndices, solutionMatrix)
+    @objc [kernel::id{MPSMatrixSolveLU} encodeToCommandBuffer:cmdbuf::id{MTLCommandBuffer}
+                                        sourceMatrix:sourceMatrix::id{MPSMatrix}
+                                        rightHandSideMatrix:rightHandSideMatrix::id{MPSMatrix}
+                                        pivotIndices:pivotIndices::id{MPSMatrix}
+                                        solutionMatrix:solutionMatrix::id{MPSMatrix}]::Nothing
+end
+
+
+export MPSMatrixSolveTriangular
+
+@objcwrapper immutable=false MPSMatrixSolveTriangular <: MPSMatrixBinaryKernel
+
+function MPSMatrixSolveTriangular(device, right, upper, transpose, unit, order, numberOfRightHandSides, alpha)
+    kernel = @objc [MPSMatrixSolveTriangular alloc]::id{MPSMatrixSolveTriangular}
+    obj = MPSMatrixSolveTriangular(kernel)
+    finalizer(release, obj)
+    @objc [obj::id{MPSMatrixSolveTriangular} initWithDevice:device::id{MTLDevice}
+                                             right:right::Bool
+                                             upper:upper::Bool
+                                             transpose:transpose::Bool
+                                             unit:unit::Bool
+                                             order:order::NSUInteger
+                                             numberOfRightHandSides:numberOfRightHandSides::NSUInteger
+                                             alpha:alpha::Cdouble]::id{MPSMatrixSolveTriangular}
+    return obj
+end
+
+function encode!(cmdbuf::MTLCommandBuffer, kernel::MPSMatrixSolveTriangular, sourceMatrix, rightHandSideMatrix, solutionMatrix)
+    @objc [kernel::id{MPSMatrixSolveTriangular} encodeToCommandBuffer:cmdbuf::id{MTLCommandBuffer}
+                                        sourceMatrix:sourceMatrix::id{MPSMatrix}
+                                        rightHandSideMatrix:rightHandSideMatrix::id{MPSMatrix}
+                                        solutionMatrix:solutionMatrix::id{MPSMatrix}]::Nothing
+end
+
+
+export MPSMatrixSolveCholesky
+
+@objcwrapper immutable=false MPSMatrixSolveCholesky <: MPSMatrixBinaryKernel
+
+function MPSMatrixSolveCholesky(device, upper, order, numberOfRightHandSides)
+    kernel = @objc [MPSMatrixSolveCholesky alloc]::id{MPSMatrixSolveCholesky}
+    obj = MPSMatrixSolveCholesky(kernel)
+    finalizer(release, obj)
+    @objc [obj::id{MPSMatrixSolveCholesky} initWithDevice:device::id{MTLDevice}
+                                           upper:upper::Bool
+                                           order:order::NSUInteger
+                                           numberOfRightHandSides:numberOfRightHandSides::NSUInteger]::id{MPSMatrixSolveCholesky}
+    return obj
+end
+
+function encode!(cmdbuf::MTLCommandBuffer, kernel::MPSMatrixSolveCholesky, sourceMatrix, rightHandSideMatrix, solutionMatrix)
+    @objc [kernel::id{MPSMatrixSolveCholesky} encodeToCommandBuffer:cmdbuf::id{MTLCommandBuffer}
+                                              sourceMatrix:sourceMatrix::id{MPSMatrix}
+                                              rightHandSideMatrix:rightHandSideMatrix::id{MPSMatrix}
+                                              solutionMatrix:solutionMatrix::id{MPSMatrix}]::Nothing
+end