Preserve aliased matrix in fallback solve

Co-Authored-By: Chris Rackauckas <accounts@chrisrackauckas.com>

Author: ChrisRackauckas

lib/NonlinearSolveBase/src/utils.jl

@@ -311,7 +311,7 @@ function linsolve_identity!!(workspace, A::AbstractMatrix)
             return LinearAlgebra.ldiv!(LinearAlgebra.LowerTriangular(A_solve), ws.rhs)
         end
     end
-    return ws.lincache(; A, b = ws.rhs).u
+    return ws.lincache(; A = A_solve, b = ws.rhs).u
 end
 
 function initial_jacobian_scaling_alpha(α, u, fu, ::Any)

lib/NonlinearSolveBase/test/linsolve_workspace.jl

@@ -40,6 +40,10 @@ end
     expected_reset = Matrix{Float64}(I, 3, 3)
     ldiv!(LowerTriangular(A_reset), expected_reset)
     @test Utils.linsolve_identity!!(workspace, A_reset) == expected_reset
+
+    A_general = [2.0 1.0 0.0; 0.0 3.0 1.0; 1.0 0.0 4.0]
+    copyto!(workspace.rhs, A_general)
+    @test Utils.linsolve_identity!!(workspace, workspace.rhs) ≈ inv(A_general)
 end
 
 @testset "singular input takes the pivoted-QR rescue" begin