أدخل القيم — يتم تحديث النتيجة فورًا. تدعم من 2×2 إلى 6×6.
A determinant is a special scalar value calculated from a square matrix. It tells you if the matrix is invertible, how it scales areas/volumes, and encodes orientation. A zero determinant means the matrix is singular (non-invertible).
صيغة لايبنيتز: det(A) = Σσ sgn(σ) Πi ai,σ(i). لأجل الكفاءة تُستخدم طرق الحذف/التحليل في الحساب العملي.
2×2: det([[a,b],[c,d]]) = ad − bc.
3×3:
4×4: استخدم التوسيع بالكوفاكتور أو LU؛ هذه الحاسبة تستخدم LU داخليًا.
A complete guide to one of the most important concepts in linear algebra
A determinant is a special scalar value that can be calculated from any square matrix (a matrix with equal rows and columns). Written as det(A) or |A|, it encapsulates fundamental properties of the matrix in a single number.
The determinant tells you whether a system of linear equations has a unique solution, whether a matrix can be inverted, and how transformations affect geometric objects like area and volume.
The absolute value of a 2×2 determinant equals the area of the parallelogram formed by the row vectors.
For 3×3 matrices, the determinant represents the volume of the parallelepiped spanned by the row vectors.
Positive determinant preserves orientation; negative means reflection. Zero means the space is "flattened."
Cramer's Rule uses determinants to solve systems of equations. If det ≠ 0, a unique solution exists.
A matrix has an inverse if and only if its determinant is non-zero. This is fundamental in linear algebra.
Finding eigenvalues requires solving det(A − λI) = 0, essential in physics and engineering.
Transformations in 3D graphics use matrices. The determinant reveals if shapes are flipped or scaled.
Watch how determinants are calculated step-by-step with interactive examples
Calculations update in real-time as you type
Support for 2×2 up to 6×6 matrices
See the calculation process explained
Copy results or matrices with one click
تقرأ هذه الحاسبة إدخالاتك وتحسب المحدد لحظيًا. وللاستقرار العددي والأداء تستخدم تحليل LU مع اختيار محوري جزئي. النتيجة تساوي حاصل ضرب قطر U مضروبًا في إشارة تبديلات الصفوف.
