# Question about singular matrix inversion

**URL:** <https://discourse.julialang.org/t/question-about-singular-matrix-inversion/126403>\
**Category:** General Usage\
**Tags:** matrix\
**Created:** [February 28, 2025, 12:07am UTC](https://discourse.julialang.org/t/question-about-singular-matrix-inversion/126403 "2025-02-28T00:07:03Z")\
**Posts on this page:** 3\
**Page:** 1

<div class="post-metadata">

**Author:** ![fdekerme](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/fdekerme/32/43574_2.png) [@fdekerme](https://discourse.julialang.org/u/fdekerme)\
**Post date:** [February 28, 2025, 12:07am UTC](https://discourse.julialang.org/t/question-about-singular-matrix-inversion/126403/1 "2025-02-28T00:07:03Z")

</div>

Hello 😀,  
I was wondering why inversion works (when it shouldn’t) with floats, but returns an error with integers? Is it a rounding issue?

```julia
julia> inv([0.23 0.23; 0.3 0.3])
2×2 Matrix{Float64}:
 -3.3777e17 2.58957e17
  3.3777e17 -2.58957e17

julia> inv([1 1; 2 2])
ERROR: SingularException(2)
Stacktrace:
  [1] checknonsingular
    @ ~/.julia/juliaup/julia-1.11.3+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/factorization.jl:69 [inlined]
  [2] _check_lu_success
    @ ~/.julia/juliaup/julia-1.11.3+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/lu.jl:84 [inlined]
  [3] #lu!#182
    @ ~/.julia/juliaup/julia-1.11.3+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/lu.jl:92 [inlined]
  [4] lu!
    @ ~/.julia/juliaup/julia-1.11.3+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/lu.jl:90 [inlined]
  [5] lu!
    @ ~/.julia/juliaup/julia-1.11.3+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/lu.jl:89 [inlined]
  [6] _lu
    @ ~/.julia/juliaup/julia-1.11.3+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/lu.jl:347 [inlined]
  [7] lu(::Matrix{Int64}; kwargs::@Kwargs{})
    @ LinearAlgebra ~/.julia/juliaup/julia-1.11.3+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/lu.jl:341
  [8] lu
    @ ~/.julia/juliaup/julia-1.11.3+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/lu.jl:341 [inlined]
  [9] inv(A::Matrix{Int64})
    @ LinearAlgebra ~/.julia/juliaup/julia-1.11.3+0.aarch64.apple.darwin14/share/julia/stdlib/v1.11/LinearAlgebra/src/dense.jl:993
 [10] top-level scope
    @ ~/Documents/julia_code/DLI_project/blend_functions.jl:313

```

Merci!,  
fdekerm

---

<div class="post-metadata">

**Author:** ![danielwe](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/danielwe/32/35657_2.png) [@danielwe](https://discourse.julialang.org/u/danielwe)\
**Post date:** [February 28, 2025, 2:21am UTC](https://discourse.julialang.org/t/question-about-singular-matrix-inversion/126403/2 "2025-02-28T02:21:42Z")

</div>

> [@fdekerme](#):
>
> Is it a rounding issue?

Correct. You can watch what happens in more detail by breaking down the inversion into it’s constituent steps. Start by computing the LU factorization of the matrix:

```julia-repl
ulia> A = [0.23 0.23; 0.3 0.3]
2×2 Matrix{Float64}:
 0.23 0.23
 0.3 0.3

julia> (; P, L, U) = lu(A)
LU{Float64, Matrix{Float64}, Vector{Int64}}
L factor:
2×2 Matrix{Float64}:
 1.0 0.0
 0.766667 1.0
U factor:
2×2 Matrix{Float64}:
 0.3 0.3
 0.0 2.96059e-18

```

Look at the last diagonal element in `U`: it should be zero, but we got `2.96e-18`. Compared to the other numbers here, `2.96e-18` is really close to zero; in fact, it’s closer to zero than the actual number hiding behind `0.3` is to the real number 0.3. (Try `big"0.3" - 0.3` to see this.) But since it’s not exactly zero, the remaining steps of the inversion will succeed and give a bogus answer:

```julia-repl
julia> Ainv = U \ (L \ P)
2×2 Matrix{Float64}:
 -3.3777e17 2.58957e17
  3.3777e17 -2.58957e17

```

The reason your second example fails as expected is not due to the number type; in fact, the type must be converted to float before the factorization can proceed anyway. The difference is simply that for these particular numbers, the algorithm used for LU factorization happens to return an exact instead of approximate zero pivot:

```julia-repl
julia> (; P, L, U) = lu([1.0 1.0; 2.0 2.0]; check=false)
Failed factorization of type LU{Float64, Matrix{Float64}, Vector{Int64}}

julia> U
2×2 Matrix{Float64}:
 2.0 2.0
 0.0 0.0

```

---

<div class="post-metadata">

**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [February 28, 2025, 12:59pm UTC](https://discourse.julialang.org/t/question-about-singular-matrix-inversion/126403/3 "2025-02-28T12:59:10Z")

</div>

> [@danielwe](#):
>
> The difference is simply that for these particular numbers, the algorithm used for LU factorization happens to return an exact instead of approximate zero pivot:

The reason for this is that `1.0` and `2.0` and `1.0/2.0` are represented exactly as floating-point numbers, so Gaussian elimination on `[1 1; 2 2]` incurs no roundoff errors.

PS. On my machine, `inv([0.23 0.23; 0.3 0.3])` happens to gives a `SingularException` too.