diff --git a/src/LinearAlgebra/Reduction/RowEchelonForm.php b/src/LinearAlgebra/Reduction/RowEchelonForm.php
index 510c6d7d7..f2be68c8d 100644
--- a/src/LinearAlgebra/Reduction/RowEchelonForm.php
+++ b/src/LinearAlgebra/Reduction/RowEchelonForm.php
@@ -127,6 +127,19 @@ public static function gaussianElimination(NumericMatrix $A): array
         $swaps = 0;
         $ε     = $A->getError();
 
+        // Scale the matrix by its smallest non-zero element
+        $min = PHP_INT_MAX;
+        for ($i = 0; $i < $m; $i++) {
+            for ($j = 0; $j < $n; $j++) {
+                $elem = \abs($A[$i][$j]);
+                if (Support::isNotZero($elem, $ε) and $elem < $min) {
+                    $min = $elem;
+                }
+            }
+        }
+
+        $R = $A->scalarMultiply(1/$min)->getMatrix();
+
         for ($k = 0; $k < $size; $k++) {
             // Find column max
             $i_max = $k;
diff --git a/tests/LinearAlgebra/Reduction/ReducedRowEchelonFormTest.php b/tests/LinearAlgebra/Reduction/ReducedRowEchelonFormTest.php
index bdb426758..240dba643 100644
--- a/tests/LinearAlgebra/Reduction/ReducedRowEchelonFormTest.php
+++ b/tests/LinearAlgebra/Reduction/ReducedRowEchelonFormTest.php
@@ -1045,6 +1045,44 @@ public function dataProviderForRref(): array
                     [0, 0, 0, 0, 0],
                 ],
             ],
+            // Large magnitude case
+            [
+                [
+                    [-2.82878905765*10**6, 0, 0, 0, 0, 0],
+                    [0, -2.82878905765*10**6, 0, 0, 0, 1729.7],
+                    [0, 0, -2.82878905765*10**6, 0, -1729.7, 0],
+                    [0, 0, 0, -2.82879105765*10**6, 0, 0],
+                    [0, 0, -1729.7, 0, -1.05765, 0],
+                    [0, 1729.7, 0, 0, 0, -1.05765]
+                ],
+                [
+                    [1, 0, 0, 0, 0, 0],
+                    [0, 1, 0, 0, 0, 0],
+                    [0, 0, 1, 0, 0, 0],
+                    [0, 0, 0, 1, 0, 0],
+                    [0, 0, 0, 0, 1, 0],
+                    [0, 0, 0, 0, 0, 1]
+                ]
+            ],
+            // Scaled version of large magnitude case
+            [
+                [
+                    [-0.99999966687087, 0, 0, 0, 0, 0],
+                    [0, -0.99999966687087, 0, 0, 0, 0.00061146285160793],
+                    [0, 0, -0.99999966687087, 0, -0.00061146285160793, 0],
+                    [0, 0, 0, -1.0000003738869, 0, 0],
+                    [0, 0, -0.00061146285160793, 0, -3.7388694340557*10**-7, 0],
+                    [0, 0.00061146285160793, 0, 0, 0, -3.7388694340557*10**-7]
+                ],
+                [
+                    [1, 0, 0, 0, 0, 0],
+                    [0, 1, 0, 0, 0, 0],
+                    [0, 0, 1, 0, 0, 0],
+                    [0, 0, 0, 1, 0, 0],
+                    [0, 0, 0, 0, 1, 0],
+                    [0, 0, 0, 0, 0, 1]
+                ]
+            ]
         ];
     }