diff --git a/README.md b/README.md
index 0af64f0..662a5eb 100644
--- a/README.md
+++ b/README.md
@@ -234,6 +234,8 @@ var error = Matrix.sub(B, A.mmul(x)); // The error enables to evaluate the solut
 ```
 #### Decompositions
 
+`QrDecomposition` along with `LuDecomposition` need a matrix with at least as many rows as columns. A matrix carrying more columns than rows is rejected with a `RangeError`.
+
 ##### QR Decomposition
 ```js
 var A = new Matrix([
diff --git a/matrix.d.ts b/matrix.d.ts
index c6f6153..ddccbb0 100644
--- a/matrix.d.ts
+++ b/matrix.d.ts
@@ -1570,6 +1570,9 @@ export class CholeskyDecomposition {
 export { CholeskyDecomposition as CHO };
 
 /**
+ * The LU decomposition of a matrix with at least as many rows as columns.
+ * A matrix carrying more columns than rows is rejected with a `RangeError`.
+ *
  * @link https://github.com/lutzroeder/Mapack/blob/master/Source/LuDecomposition.cs
  */
 export class LuDecomposition {
@@ -1585,6 +1588,9 @@ export class LuDecomposition {
 export { LuDecomposition as LU };
 
 /**
+ * The QR decomposition of a matrix with at least as many rows as columns.
+ * A matrix carrying more columns than rows is rejected with a `RangeError`.
+ *
  * @link https://github.com/lutzroeder/Mapack/blob/master/Source/QrDecomposition.cs
  */
 export class QrDecomposition {
diff --git a/src/__tests__/decompositions/lu.test.js b/src/__tests__/decompositions/lu.test.js
index 39b2298..ba525ec 100644
--- a/src/__tests__/decompositions/lu.test.js
+++ b/src/__tests__/decompositions/lu.test.js
@@ -45,6 +45,15 @@ describe('LU decomposition', () => {
         new LU([
           [0, 1, 2],
           [0, 1, 2],
+        ]),
+    ).toThrow('Matrix must have at least as many rows as columns');
+
+    expect(
+      () =>
+        new LU([
+          [0, 1],
+          [0, 1],
+          [0, 1],
         ]).determinant,
     ).toThrow('Matrix must be square');
   });
diff --git a/src/__tests__/decompositions/shape.test.js b/src/__tests__/decompositions/shape.test.js
new file mode 100644
index 0000000..1b98ee2
--- /dev/null
+++ b/src/__tests__/decompositions/shape.test.js
@@ -0,0 +1,74 @@
+import { describe, it, expect } from 'vitest';
+
+import { Matrix, LuDecomposition, QrDecomposition, solve } from '../..';
+
+const message = /^Matrix must have at least as many rows as columns$/;
+
+describe('LU and QR need at least as many rows as columns', () => {
+  const wide = new Matrix([
+    [1, 2, 3],
+    [4, 5, 6],
+  ]);
+
+  it('LU rejects a wide matrix', () => {
+    expect(() => new LuDecomposition(wide)).toThrow(message);
+  });
+
+  it('QR rejects a wide matrix', () => {
+    expect(() => new QrDecomposition(wide)).toThrow(message);
+  });
+
+  it('LU rejects a wide 2D array', () => {
+    expect(() => new LuDecomposition([[1, 2, 3, 4]])).toThrow(message);
+  });
+
+  it('QR rejects a wide 2D array', () => {
+    expect(() => new QrDecomposition([[1, 2, 3, 4]])).toThrow(message);
+  });
+
+  it('solve reports the shape rather than a rank problem', () => {
+    expect(() => solve(wide, Matrix.columnVector([1, 2]))).toThrow(message);
+  });
+});
+
+describe('LU and QR keep working on the supported shapes', () => {
+  const tall = new Matrix([
+    [1, 2],
+    [3, 4],
+    [5, 6],
+  ]);
+  const square = new Matrix([
+    [4, 3],
+    [6, 3],
+  ]);
+
+  it('QR on a tall matrix rebuilds the input', () => {
+    const qr = new QrDecomposition(tall);
+    const product = qr.orthogonalMatrix.mmul(qr.upperTriangularMatrix);
+    for (let i = 0; i < tall.rows; i++) {
+      for (let j = 0; j < tall.columns; j++) {
+        expect(product.get(i, j)).toBeCloseTo(tall.get(i, j), 10);
+      }
+    }
+  });
+
+  it('LU on a tall matrix reports a usable triangular pair', () => {
+    const lu = new LuDecomposition(tall);
+    expect(lu.lowerTriangularMatrix.rows).toBe(3);
+    expect(lu.upperTriangularMatrix.columns).toBe(2);
+  });
+
+  it('LU on a square matrix stays solvable', () => {
+    const lu = new LuDecomposition(square);
+    expect(lu.isSingular()).toBe(false);
+    const x = lu.solve(Matrix.columnVector([10, 12]));
+    const back = square.mmul(x);
+    expect(back.get(0, 0)).toBeCloseTo(10, 10);
+    expect(back.get(1, 0)).toBeCloseTo(12, 10);
+  });
+
+  it('an empty matrix is still accepted', () => {
+    expect(() => new LuDecomposition(new Matrix(0, 0))).not.toThrow();
+    expect(() => new QrDecomposition(new Matrix(0, 0))).not.toThrow();
+  });
+});
diff --git a/src/dc/lu.js b/src/dc/lu.js
index 68b12b0..227cada 100644
--- a/src/dc/lu.js
+++ b/src/dc/lu.js
@@ -4,6 +4,9 @@ import WrapperMatrix2D from '../wrap/WrapperMatrix2D';
 export default class LuDecomposition {
   constructor(matrix) {
     matrix = WrapperMatrix2D.checkMatrix(matrix);
+    if (matrix.rows < matrix.columns) {
+      throw new RangeError('Matrix must have at least as many rows as columns');
+    }
 
     let lu = matrix.clone();
     let rows = lu.rows;
diff --git a/src/dc/qr.js b/src/dc/qr.js
index ffce782..f3d7425 100644
--- a/src/dc/qr.js
+++ b/src/dc/qr.js
@@ -6,6 +6,9 @@ import { hypotenuse } from './util';
 export default class QrDecomposition {
   constructor(value) {
     value = WrapperMatrix2D.checkMatrix(value);
+    if (value.rows < value.columns) {
+      throw new RangeError('Matrix must have at least as many rows as columns');
+    }
 
     let qr = value.clone();
     let m = value.rows;