diff --git a/tests/FsMath.Tests/FsMath.Tests.fsproj b/tests/FsMath.Tests/FsMath.Tests.fsproj
index d6a61c2..9551d2b 100644
--- a/tests/FsMath.Tests/FsMath.Tests.fsproj
+++ b/tests/FsMath.Tests/FsMath.Tests.fsproj
@@ -32,6 +32,7 @@
     <Compile Include="LinearAlgebraTests.fs" />
     <Compile Include="LinearAlgebraCoverageTests.fs" />
     <Compile Include="LinearAlgebraAdvancedTests.fs" />
+    <Compile Include="LinearAlgebraErrorTests.fs" />
     <Compile Include="EVDTests.fs" />
     <Compile Include="SVDTests.fs" />
     <Compile Include="BidiagonalizationCoverageTests.fs" />
diff --git a/tests/FsMath.Tests/LinearAlgebraErrorTests.fs b/tests/FsMath.Tests/LinearAlgebraErrorTests.fs
new file mode 100644
index 0000000..bdf4119
--- /dev/null
+++ b/tests/FsMath.Tests/LinearAlgebraErrorTests.fs
@@ -0,0 +1,155 @@
+namespace FsMath.Tests.LinearAlgebraErrorTests
+
+open System
+open Xunit
+open FsMath
+open FsMath.Algebra
+
+open FsMath.Tests
+open FsMath.Tests.AssertHelpers
+
+open FsMath.Tests.ExpectoStyle
+
+/// Tests for error handling paths in LinearAlgebra module
+/// These tests target previously uncovered exception paths
+
+module BackSubstituteErrorTests =
+
+    [<Fact>]
+    let ``Throws ArgumentException on zero diagonal during back substitution`` () =
+        // Create an upper triangular matrix with zero on diagonal at position [1,1]
+        // This tests line 110 in LinearAlgebra.fs
+        let r = Matrix(3, 3, [| 2.0; 3.0; 4.0;
+                               0.0; 0.0; 5.0;   // Zero at r[1,1]
+                               0.0; 0.0; 6.0 |])
+        let y = [| 10.0; 15.0; 18.0 |]
+        throws<ArgumentException>(fun () -> LinearAlgebra.backSubstitute r y |> ignore) "should throw on zero diagonal"
+
+
+module SolveTriangularLinearSystemErrorTests =
+
+    [<Fact>]
+    let ``Throws ArgumentException on zero diagonal in forward substitution (lower triangular)`` () =
+        // Test error path at line 281 in LinearAlgebra.fs
+        // Create lower triangular with zero diagonal at L[1,1]
+        let L = Matrix(3, 3, [| 1.0; 0.0; 0.0;
+                               2.0; 0.0; 0.0;   // Zero at L[1,1]
+                               3.0; 4.0; 1.0 |])
+        let b = [| 2.0; 5.0; 20.0 |]
+        throws<ArgumentException>(fun () -> LinearAlgebra.solveTriangularLinearSystem L b true |> ignore) "should throw on zero diagonal in forward substitution"
+
+    [<Fact>]
+    let ``Throws ArgumentException on zero diagonal in backward substitution (upper triangular)`` () =
+        // Test error path at line 293 in LinearAlgebra.fs
+        // Create upper triangular with zero diagonal at U[1,1]
+        let U = Matrix(3, 3, [| 1.0; 2.0; 3.0;
+                               0.0; 0.0; 5.0;   // Zero at U[1,1]
+                               0.0; 0.0; 6.0 |])
+        let b = [| 23.0; 9.0; 6.0 |]
+        throws<ArgumentException>(fun () -> LinearAlgebra.solveTriangularLinearSystem U b false |> ignore) "should throw on zero diagonal in backward substitution"
+
+    [<Fact>]
+    let ``Throws ArgumentException on zero diagonal at last position (backward)`` () =
+        // Test zero diagonal in last position during backward substitution
+        let U = Matrix(2, 2, [| 2.0; 3.0;
+                               0.0; 0.0 |])   // Zero at U[1,1]
+        let b = [| 10.0; 5.0 |]
+        throws<ArgumentException>(fun () -> LinearAlgebra.solveTriangularLinearSystem U b false |> ignore) "should throw on zero diagonal at end"
+
+    [<Fact>]
+    let ``Throws ArgumentException on zero diagonal at first position (forward)`` () =
+        // Test zero diagonal in first position during forward substitution
+        let L = Matrix(2, 2, [| 0.0; 0.0;       // Zero at L[0,0]
+                               1.0; 2.0 |])
+        let b = [| 1.0; 5.0 |]
+        throws<ArgumentException>(fun () -> LinearAlgebra.solveTriangularLinearSystem L b true |> ignore) "should throw on zero diagonal at start"
+
+
+module CholeskyErrorTests =
+
+    [<Fact>]
+    let ``Throws ArgumentException on non-square matrix for cholesky`` () =
+        // Test error path at line 466 in LinearAlgebra.fs
+        let A = Matrix(3, 2, [| 4.0; 2.0;
+                               2.0; 3.0;
+                               1.0; 1.0 |])
+        throws<ArgumentException>(fun () -> LinearAlgebra.cholesky A |> ignore) "should throw on non-square matrix"
+
+    [<Fact>]
+    let ``Throws ArgumentException on 4x2 rectangular matrix`` () =
+        // Additional test for non-square error path
+        let A = Matrix(4, 2, [| 1.0; 2.0;
+                               3.0; 4.0;
+                               5.0; 6.0;
+                               7.0; 8.0 |])
+        throws<ArgumentException>(fun () -> LinearAlgebra.cholesky A |> ignore) "should throw on 4x2 matrix"
+
+    [<Fact>]
+    let ``Throws ArgumentException on 2x4 rectangular matrix`` () =
+        // Test with more columns than rows
+        let A = Matrix(2, 4, [| 1.0; 2.0; 3.0; 4.0;
+                               5.0; 6.0; 7.0; 8.0 |])
+        throws<ArgumentException>(fun () -> LinearAlgebra.cholesky A |> ignore) "should throw on 2x4 matrix"
+
+
+module SolveLinearSystemsErrorTests =
+
+    [<Fact>]
+    let ``Throws ArgumentException when B rows don't match A rows in solveLinearSystems`` () =
+        // Test error path at line 636 in LinearAlgebra.fs
+        let A = Matrix.identity 3
+        let B = Matrix(2, 2, [| 1.0; 2.0;
+                               3.0; 4.0 |])  // Only 2 rows, but A has 3
+        throws<ArgumentException>(fun () -> LinearAlgebra.solveLinearSystems A B |> ignore) "should throw on row mismatch"
+
+    [<Fact>]
+    let ``Throws ArgumentException when B has fewer rows than A`` () =
+        // Additional test for row mismatch
+        let A = Matrix(4, 4, Array.init 16 (fun i -> float (i + 1)))
+        let B = Matrix(3, 2, [| 1.0; 2.0;
+                               3.0; 4.0;
+                               5.0; 6.0 |])  // 3 rows vs A's 4 rows
+        throws<ArgumentException>(fun () -> LinearAlgebra.solveLinearSystems A B |> ignore) "should throw when B has fewer rows"
+
+    [<Fact>]
+    let ``Throws ArgumentException when B has more rows than A`` () =
+        // Test opposite direction
+        let A = Matrix(2, 2, [| 1.0; 2.0;
+                               3.0; 4.0 |])
+        let B = Matrix(3, 2, [| 1.0; 2.0;
+                               3.0; 4.0;
+                               5.0; 6.0 |])  // 3 rows vs A's 2 rows
+        throws<ArgumentException>(fun () -> LinearAlgebra.solveLinearSystems A B |> ignore) "should throw when B has more rows"
+
+
+module SolveLinearSystemErrorTests =
+
+    [<Fact>]
+    let ``Throws ArgumentException when vector b length doesn't match A rows in solveLinearSystem`` () =
+        // Test error path at line 661 in LinearAlgebra.fs
+        let A = Matrix.identity 3
+        let b = [| 1.0; 2.0 |]  // Only 2 elements, but A has 3 rows
+        throws<ArgumentException>(fun () -> LinearAlgebra.solveLinearSystem A b |> ignore) "should throw on dimension mismatch"
+
+    [<Fact>]
+    let ``Throws ArgumentException when vector b is shorter than A rows`` () =
+        // Additional test
+        let A = Matrix(4, 4, Array.init 16 (fun i -> float (i + 1)))
+        let b = [| 1.0; 2.0; 3.0 |]  // 3 elements vs A's 4 rows
+        throws<ArgumentException>(fun () -> LinearAlgebra.solveLinearSystem A b |> ignore) "should throw when b is shorter"
+
+    [<Fact>]
+    let ``Throws ArgumentException when vector b is longer than A rows`` () =
+        // Test opposite direction
+        let A = Matrix(2, 2, [| 1.0; 2.0;
+                               3.0; 4.0 |])
+        let b = [| 1.0; 2.0; 3.0 |]  // 3 elements vs A's 2 rows
+        throws<ArgumentException>(fun () -> LinearAlgebra.solveLinearSystem A b |> ignore) "should throw when b is longer"
+
+    [<Fact>]
+    let ``Throws ArgumentException with empty vector on non-empty matrix`` () =
+        // Edge case: empty vector
+        let A = Matrix(2, 2, [| 1.0; 2.0;
+                               3.0; 4.0 |])
+        let b = [||]  // Empty vector
+        throws<ArgumentException>(fun () -> LinearAlgebra.solveLinearSystem A b |> ignore) "should throw on empty vector"