diff --git a/src/generic.jl b/src/generic.jl
index c8336f87..83c2c1c8 100644
--- a/src/generic.jl
+++ b/src/generic.jl
@@ -1236,14 +1236,22 @@ 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
+        istrium1 = istriu(A, -1)
+        istril1 = istril(A, 1)
+        if istril1 && iszero(diagview(A,1)) # istril(A)
+            if istrium1
+                if iszero(diagview(A, -1))
+                    return Diagonal(A) \ B
+                end
+                return Bidiagonal(A, :L) \ B
             else
                 return LowerTriangular(A) \ B
             end
         end
-        if istriu(A)
+        if istrium1 && iszero(diagview(A, -1)) # istriu(A)
+            if istril1
+                return Bidiagonal(A, :U) \ B
+            end
             return UpperTriangular(A) \ B
         end
         return lu(A) \ B
diff --git a/test/generic.jl b/test/generic.jl
index 521beaf4..59760db0 100644
--- a/test/generic.jl
+++ b/test/generic.jl
@@ -937,4 +937,16 @@ end
     @test B == A2
 end
 
+@testset "linear solve for dense banded matrices" begin
+    b = randn(5)
+    for A in (diagm(0=>1:5, -1=>1:4, 1=>1:4), # SymTridiagonal
+              diagm(0=>1:5, -1=>1:4, 1=>5:8), # Tridiagonal
+              diagm(0=>1:5, 1=>1:4), # Upper Bidiagonal
+              diagm(0=>1:5, -1=>1:4), # Lower Bidiagonal
+            )
+        x = A \ b
+        @test A * x ≈ b
+    end
+end
+
 end # module TestGeneric