diff --git a/tests/FsMath.Tests/FsMath.Tests.fsproj b/tests/FsMath.Tests/FsMath.Tests.fsproj
index bce3ed5..85ae95f 100644
--- a/tests/FsMath.Tests/FsMath.Tests.fsproj
+++ b/tests/FsMath.Tests/FsMath.Tests.fsproj
@@ -34,6 +34,7 @@
     <Compile Include="LinearAlgebraCoverageTests.fs" />
     <Compile Include="LinearAlgebraAdvancedTests.fs" />
     <Compile Include="LinearAlgebraErrorTests.fs" />
+    <Compile Include="LinearAlgebraErrorTestsAdditional.fs" />
     <Compile Include="EVDTests.fs" />
     <Compile Include="SVDTests.fs" />
     <Compile Include="BidiagonalizationCoverageTests.fs" />
diff --git a/tests/FsMath.Tests/LinearAlgebraErrorTestsAdditional.fs b/tests/FsMath.Tests/LinearAlgebraErrorTestsAdditional.fs
new file mode 100644
index 0000000..ab9e8c6
--- /dev/null
+++ b/tests/FsMath.Tests/LinearAlgebraErrorTestsAdditional.fs
@@ -0,0 +1,222 @@
+namespace FsMath.Tests.LinearAlgebraErrorTestsAdditional
+
+open System
+open Xunit
+open FsMath
+open FsMath.Algebra
+
+open FsMath.Tests
+open FsMath.Tests.AssertHelpers
+open FsMath.Tests.ExpectoStyle
+
+/// Additional comprehensive error tests for LinearAlgebra module
+/// Targeting specific uncovered error paths with multiple test variations
+
+module BackSubstituteZeroDiagonalTests =
+
+    [<Fact>]
+    let ``backSubstitute throws on zero diagonal at position 0`` () =
+        // Line 110: Test zero diagonal at first position
+        let r = Matrix(1, 1, [| 0.0 |])
+        let y = [| 5.0 |]
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.backSubstitute r y |> ignore)
+        Assert.Contains("Diagonal element is zero", ex.Message)
+
+    [<Fact>]
+    let ``backSubstitute throws on zero diagonal in middle`` () =
+        // Line 110: Test zero diagonal in the middle of matrix
+        let r = Matrix(4, 4, [|
+            5.0; 1.0; 2.0; 3.0;
+            0.0; 0.0; 1.0; 2.0;  // Zero at r[1,1]
+            0.0; 0.0; 4.0; 1.0;
+            0.0; 0.0; 0.0; 3.0 |])
+        let y = [| 10.0; 5.0; 8.0; 9.0 |]
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.backSubstitute r y |> ignore)
+        Assert.Contains("Diagonal element is zero", ex.Message)
+
+    [<Fact>]
+    let ``backSubstitute throws on zero diagonal at last position`` () =
+        // Line 110: Test zero diagonal at last position
+        let r = Matrix(3, 3, [|
+            2.0; 3.0; 4.0;
+            0.0; 5.0; 1.0;
+            0.0; 0.0; 0.0 |])  // Zero at r[2,2]
+        let y = [| 10.0; 15.0; 18.0 |]
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.backSubstitute r y |> ignore)
+        Assert.Contains("Diagonal element is zero", ex.Message)
+
+
+module SolveTriangularSystemZeroDiagonalTests =
+
+    [<Fact>]
+    let ``solveTriangularLinearSystem throws on zero diagonal - forward sub at position 0`` () =
+        // Line 281: Zero diagonal at start in forward substitution
+        let L = Matrix(3, 3, [|
+            0.0; 0.0; 0.0;       // Zero at L[0,0]
+            1.0; 2.0; 0.0;
+            2.0; 3.0; 4.0 |])
+        let v = [| 1.0; 5.0; 10.0 |]
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.solveTriangularLinearSystem L v true |> ignore)
+        Assert.Contains("Diagonal element is zero", ex.Message)
+        Assert.Contains("K[0,0]", ex.Message)
+
+    [<Fact>]
+    let ``solveTriangularLinearSystem throws on zero diagonal - forward sub in middle`` () =
+        // Line 281: Zero diagonal in middle position
+        let L = Matrix(4, 4, [|
+            1.0; 0.0; 0.0; 0.0;
+            2.0; 3.0; 0.0; 0.0;
+            1.0; 4.0; 0.0; 0.0;  // Zero at L[2,2]
+            3.0; 2.0; 1.0; 5.0 |])
+        let v = [| 2.0; 8.0; 10.0; 15.0 |]
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.solveTriangularLinearSystem L v true |> ignore)
+        Assert.Contains("Diagonal element is zero", ex.Message)
+        Assert.Contains("K[2,2]", ex.Message)
+
+    [<Fact>]
+    let ``solveTriangularLinearSystem throws on zero diagonal - backward sub at position 0`` () =
+        // Line 293: Zero diagonal at start in backward substitution
+        let U = Matrix(3, 3, [|
+            0.0; 1.0; 2.0;       // Zero at U[0,0]
+            0.0; 3.0; 4.0;
+            0.0; 0.0; 5.0 |])
+        let v = [| 10.0; 12.0; 15.0 |]
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.solveTriangularLinearSystem U v false |> ignore)
+        Assert.Contains("Diagonal element is zero", ex.Message)
+        Assert.Contains("K[0,0]", ex.Message)
+
+    [<Fact>]
+    let ``solveTriangularLinearSystem throws on zero diagonal - backward sub in middle`` () =
+        // Line 293: Zero diagonal in middle position
+        let U = Matrix(4, 4, [|
+            5.0; 1.0; 2.0; 3.0;
+            0.0; 0.0; 4.0; 1.0;  // Zero at U[1,1]
+            0.0; 0.0; 3.0; 2.0;
+            0.0; 0.0; 0.0; 4.0 |])
+        let v = [| 20.0; 15.0; 12.0; 8.0 |]
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.solveTriangularLinearSystem U v false |> ignore)
+        Assert.Contains("Diagonal element is zero", ex.Message)
+        Assert.Contains("K[1,1]", ex.Message)
+
+    [<Fact>]
+    let ``solveTriangularLinearSystem throws on zero diagonal - backward sub at last`` () =
+        // Line 293: Zero diagonal at last position
+        let U = Matrix(2, 2, [|
+            5.0; 2.0;
+            0.0; 0.0 |])         // Zero at U[1,1]
+        let v = [| 10.0; 5.0 |]
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.solveTriangularLinearSystem U v false |> ignore)
+        Assert.Contains("Diagonal element is zero", ex.Message)
+        Assert.Contains("K[1,1]", ex.Message)
+
+
+module CholeskyNonSquareTests =
+
+    [<Fact>]
+    let ``cholesky throws on 1x2 matrix`` () =
+        // Line 466: Non-square matrix (more columns than rows)
+        let A = Matrix(1, 2, [| 1.0; 2.0 |])
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.cholesky A |> ignore)
+        Assert.Contains("matrix must be square", ex.Message)
+
+    [<Fact>]
+    let ``cholesky throws on 2x1 matrix`` () =
+        // Line 466: Non-square matrix (more rows than columns)
+        let A = Matrix(2, 1, [| 1.0; 2.0 |])
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.cholesky A |> ignore)
+        Assert.Contains("matrix must be square", ex.Message)
+
+    [<Fact>]
+    let ``cholesky throws on 5x3 matrix`` () =
+        // Line 466: Larger non-square matrix
+        let A = Matrix(5, 3, Array.init 15 (fun i -> float (i + 1)))
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.cholesky A |> ignore)
+        Assert.Contains("matrix must be square", ex.Message)
+
+    [<Fact>]
+    let ``cholesky throws on 3x7 matrix`` () =
+        // Line 466: Wide non-square matrix
+        let A = Matrix(3, 7, Array.init 21 (fun i -> float (i + 1)))
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.cholesky A |> ignore)
+        Assert.Contains("matrix must be square", ex.Message)
+
+
+module SolveLinearSystemsRowMismatchTests =
+
+    [<Fact>]
+    let ``solveLinearSystems throws when B has 1 row and A has 2 rows`` () =
+        // Line 636: B rows don't match A rows
+        let A = Matrix.identity 2
+        let B = Matrix(1, 1, [| 1.0 |])
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.solveLinearSystems A B |> ignore)
+        Assert.Contains("same number of rows", ex.Message)
+
+    [<Fact>]
+    let ``solveLinearSystems throws when B has 5 rows and A has 3 rows`` () =
+        // Line 636: B has more rows than A
+        let A = Matrix(3, 3, Array.init 9 (fun i -> float (i + 1)))
+        let B = Matrix(5, 2, Array.init 10 (fun i -> float (i + 1)))
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.solveLinearSystems A B |> ignore)
+        Assert.Contains("same number of rows", ex.Message)
+
+    [<Fact>]
+    let ``solveLinearSystems throws when B has 2 rows and A has 4 rows`` () =
+        // Line 636: B has fewer rows than A
+        let A = Matrix(4, 4, Array.init 16 (fun i -> float (i + 1)))
+        let B = Matrix(2, 3, Array.init 6 (fun i -> float (i + 1)))
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.solveLinearSystems A B |> ignore)
+        Assert.Contains("same number of rows", ex.Message)
+
+
+module SolveLinearSystemVectorMismatchTests =
+
+    [<Fact>]
+    let ``solveLinearSystem throws when b has 1 element and A has 2 rows`` () =
+        // Line 661: Vector length doesn't match matrix rows
+        let A = Matrix.identity 2
+        let b = [| 1.0 |]
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.solveLinearSystem A b |> ignore)
+        Assert.Contains("length = A.NumRows", ex.Message)
+
+    [<Fact>]
+    let ``solveLinearSystem throws when b has 5 elements and A has 3 rows`` () =
+        // Line 661: Vector longer than matrix rows
+        let A = Matrix(3, 3, Array.init 9 (fun i -> float (i + 1)))
+        let b = [| 1.0; 2.0; 3.0; 4.0; 5.0 |]
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.solveLinearSystem A b |> ignore)
+        Assert.Contains("length = A.NumRows", ex.Message)
+
+    [<Fact>]
+    let ``solveLinearSystem throws when b has 2 elements and A has 5 rows`` () =
+        // Line 661: Vector shorter than matrix rows
+        let A = Matrix(5, 5, Array.init 25 (fun i -> float (i + 1)))
+        let b = [| 1.0; 2.0 |]
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.solveLinearSystem A b |> ignore)
+        Assert.Contains("length = A.NumRows", ex.Message)
+
+    [<Fact>]
+    let ``solveLinearSystem throws on completely empty vector with 1x1 matrix`` () =
+        // Line 661: Empty vector edge case
+        let A = Matrix(1, 1, [| 5.0 |])
+        let b = [||]
+        let ex = Assert.Throws<ArgumentException>(fun () ->
+            LinearAlgebra.solveLinearSystem A b |> ignore)
+        Assert.Contains("length = A.NumRows", ex.Message)