Calculatrice de déterminant
Cette calculatrice trouve exactement le déterminant d’une matrice 2×2, 3×3 ou 4×4. Elle montre le développement par cofacteurs étape par étape et vérifie la réponse d’une seconde façon, par réduction de lignes.
Résultat
Saisissez les valeurs et appuyez sur Calculer pour voir le résultat.
Formule
A 2×2 determinant: multiply down the main diagonal, subtract the other diagonal.
A 3×3 determinant by cofactor expansion along the first row, with the signs + − +. A 4×4 works the same way using four 3×3 minors.
Comment utiliser cette calculatrice
- Choose the matrix size: 2 × 2, 3 × 3 or 4 × 4.
- Fill in the grid row by row. Whole numbers, decimals and fractions such as 3/4 all work.
- Press Calculate. You get the determinant, whether the matrix is invertible, and every step of the cofactor expansion.
Exemples corrigés
Easy: a 2 × 2 matrix
For [[3, 8], [4, 6]], use ad − bc.
3 · 6 − 8 · 4 = 18 − 32 = −14.
The determinant is not 0, so the matrix is invertible.
Medium: a 3 × 3 matrix
Expand along the first row (6, 1, 1) with signs + − +.
Minors: (−2)(7) − 5 · 8 = −54, 4 · 7 − 5 · 2 = 18, and 4 · 8 − (−2)(2) = 36.
det = 6(−54) − 1(18) + 1(36) = −324 − 18 + 36 = −306.
Hard: a 4 × 4 matrix
Column 2 has only one non-zero entry (the 1 in row 3), so expanding down that column is quickest by hand.
Its sign is (−1)³⁺² = −1, and the 3 × 3 minor left after removing row 3 and column 2 is [[1, 2, −1], [3, 0, 5], [1, 5, 0]], whose determinant is −25 + 10 − 15 = −30.
det = −1 × 1 × (−30) = 30. The calculator expands along the first row instead and gets the same 30, then confirms it by row reduction.
Erreurs fréquentes
- Getting the signs wrong in a cofactor expansion: along the first row they go + − + −.
- Subtracting the diagonals the wrong way round: for a 2×2 it is ad − bc, not bc − ad.
- Using the wrong minor: delete the whole row and column of the entry you are expanding on.
- Slips with negative entries: (−2)(7) is −14, and subtracting a negative minor adds.
Questions fréquentes
What is the formula for a 3×3 determinant?
a(ei − fh) − b(di − fg) + c(dh − eg), where the first row is a, b, c, the second d, e, f and the third g, h, i. Each bracket is a 2×2 determinant (a minor).
What does a determinant of 0 mean?
The matrix is singular: it has no inverse, its rows (and columns) are linearly dependent, and a system of equations with this coefficient matrix has either no solution or infinitely many.
Can the determinant calculator do 4×4 matrices?
Yes — 2×2, 3×3 and 4×4. A 4×4 determinant is expanded into four 3×3 minors, and each minor’s value is shown.
Can I enter fractions or decimals?
Yes. Entries such as 3/4, −2/5 or 0.5 are kept exact, so the determinant is exact too (with a decimal shown when it is a fraction).
How is the answer checked?
Two different methods: cofactor expansion (shown in the steps) and row reduction to triangular form. The result is marked verified only when both agree.