Calculateur de Matrices
Additionnez, soustrayez et multipliez des matrices, et calculez le déterminant, l'inverse et la transposée
Matrix Calculator
Calculate the determinant of a 2x2 matrix
Enter matrix values [a b; c d]
det = ad - bcWhat is a Matrix Calculator?
A Matrix Calculator is a math tool for working with matrices—rectangular grids of numbers arranged in rows and columns. Matrices are used to represent and solve problems involving systems of equations, transformations, and data organized in table form. They are foundational in algebra, calculus, statistics, engineering, physics, computer graphics, machine learning, and many other fields.
Common matrix operations include addition, subtraction, multiplication, transpose, determinant, and inverse. Doing these operations by hand can be time-consuming and easy to get wrong, especially for 3×3 or larger matrices. A matrix calculator helps you compute results instantly and accurately.
Common Matrix Operations
- Add (A + B) -- add corresponding entries
- Subtract (A − B) -- subtract corresponding entries
- Multiply (A × B) -- row-by-column dot products
- Transpose (Aᵀ) -- flip rows and columns
- Determinant (det(A)) -- a scalar that describes matrix properties
- Inverse (A⁻¹) -- the matrix that undoes A
Matrices are especially important for solving multiple linear equations at once, transforming coordinates in 2D/3D graphics (rotation, scaling), modeling networks and relationships (graphs, Markov chains), and representing datasets and computations in engineering and science.
How to Use This Matrix Calculator
- Choose the matrix size -- for example 2×2, 3×3, etc., if the calculator supports sizing
- Enter the matrix values -- fill in the grid (Matrix A, and Matrix B if needed)
- Select the operation -- such as add, subtract, multiply, transpose, determinant, or inverse
- Click "Calculate" -- to generate the result
- Review the output -- confirm the dimensions match what you expect
Tips:
- Addition / Subtraction: matrices must be the same size
- Multiplication: the number of columns in A must equal the number of rows in B
- Inverse: only square matrices (like 2×2, 3×3) can have an inverse, and only if the determinant is not zero
Matrix Formulas
Addition / Subtraction
For matrices A and B of the same size:
(A ± B)ᵢⱼ = Aᵢⱼ ± Bᵢⱼ
Add or subtract corresponding entries
Transpose
Flip rows and columns:
(Aᵀ)ᵢⱼ = Aⱼᵢ
Row i becomes column i
Matrix Multiplication
If A is m×n and B is n×p, then A×B is m×p:
(AB)ᵢⱼ = Σ Aᵢₖ × Bₖⱼ (k = 1 to n)
Each entry is a dot product of a row of A and a column of B
Determinant (2×2)
For A = [a b; c d]:
det(A) = ad − bc
Larger matrices use cofactor expansion
Inverse (2×2)
If det(A) ≠ 0:
A⁻¹ = (1/det(A)) × [d, −b; −c, a]
3×3+ uses cofactors or row reduction
Example Calculations
Example 1: Matrix Addition
A = [1 2; 3 4], B = [5 6; 7 8]
Calculation: add entry-by-entry
Result: [6 8; 10 12]
Example 2: Matrix Multiplication
A = [1 2; 3 4], B = [2 0; 1 2]
Calculation:
- (1,1): 1×2 + 2×1 = 4
- (1,2): 1×0 + 2×2 = 4
- (2,1): 3×2 + 4×1 = 10
- (2,2): 3×0 + 4×2 = 8
Result: [4 4; 10 8]
Example 3: Transpose
A = [1 2 3; 4 5 6] (2×3 matrix)
Transpose: flip rows and columns
Result: Aᵀ = [1 4; 2 5; 3 6] (3×2 matrix)
Example 4: Determinant and Inverse (2×2)
A = [4 7; 2 6]
Determinant: 4×6 − 7×2 = 24 − 14 = 10
Since det(A) ≠ 0, inverse exists:
A⁻¹ = (1/10) × [6, −7; −2, 4] = [0.6, −0.7; −0.2, 0.4]
Frequently Asked Questions
What is a matrix used for?
Matrices represent structured data and transformations. They're used to solve systems of linear equations, perform coordinate transformations in graphics, model networks, and handle computations in science and engineering.
When can I add or subtract matrices?
Only when they have the same dimensions (same number of rows and columns). You add or subtract corresponding entries.
When can I multiply matrices?
Matrix multiplication requires that the number of columns in A equals the number of rows in B. If A is m×n, B must be n×p.
What does the determinant tell me?
The determinant is a single number that indicates properties of a square matrix. If det(A) = 0, the matrix is singular (not invertible). If det(A) is not zero, an inverse exists.
Why doesn't my matrix have an inverse?
A matrix must be square and have a non-zero determinant to be invertible. If the determinant is 0, the matrix has no inverse (it is called singular).
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Qu'est-ce qu'une matrice ?
Une matrice est un tableau rectangulaire de nombres organisés en lignes et en colonnes. Les matrices sont au cœur de l'algèbre linéaire et interviennent en infographie, en ingénierie, en économie, en apprentissage automatique et dans la résolution de systèmes d'équations. Qu'il s'agisse de faire pivoter un objet 3D ou d'entraîner un réseau de neurones, les matrices sont le langage mathématique sous-jacent.
Ce calculateur prend en charge les opérations les plus courantes sur les matrices 2×2 et 3×3 : addition, soustraction, multiplication scalaire, produit matriciel, déterminant, inverse et transposée. Saisissez vos valeurs, choisissez l'opération et obtenez le résultat immédiatement, sans calcul à la main.
Comment utiliser le calculateur de matrices
- Sélectionnez l'opération souhaitée dans le menu déroulant.
- Saisissez les valeurs de la matrice dans la grille (ligne par ligne, de gauche à droite).
- Pour les opérations à deux matrices (addition, soustraction, multiplication), remplissez la Matrice A et la Matrice B.
- Cliquez sur Calculer et lisez le résultat ci-dessous.
Formules essentielles
Déterminant 2×2 :
|A| = ad − bc pour A = [[a,b],[c,d]]
Inverse 2×2 :
A⁻¹ = (1/|A|) × [[d,−b],[−c,a]]
Produit matriciel (A×B) :
C[i][j] = Σ A[i][k] × B[k][j]
Transposée : (Aᵀ)[i][j] = A[j][i]La multiplication matricielle n'est PAS commutative — A×B est généralement différent de B×A. Une matrice n'a pas d'inverse lorsque son déterminant est nul (matrice singulière).
Exemples résolus
Déterminant 2×2
Pour A = [[3, 8], [4, 6]] : |A| = (3×6) − (8×4) = 18 − 32 = −14.
Addition de matrices
A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]] : A + B = [[6, 8], [10, 12]].
Transposée
Pour A = [[1, 2, 3], [4, 5, 6]] : Aᵀ = [[1, 4], [2, 5], [3, 6]]. Les lignes deviennent des colonnes.