📏 Matrix Determinant Calculator
Calculate the determinant of a square matrix
How to Use This Calculator
Select Matrix Size
Choose the size of your square matrix (2×2, 3×3, or 4×4).
Enter Matrix Elements
Input all elements of your square matrix.
Calculate
Click to compute the determinant.
Interpret Result
If det = 0, matrix is singular (non-invertible). Otherwise, it's invertible.
Formula
det(A) = Σ (-1)^(i+j) × aᵢⱼ × det(Mᵢⱼ)
Laplace expansion using cofactors
2×2 Matrix:
det = ad - bc
For matrix [[a b], [c d]]
3×3 Matrix:
det = a(ei − fh) − b(di − fg) + c(dh − eg)
For matrix [[a b c], [d e f], [g h i]]
Properties:
- det(AB) = det(A) × det(B)
- det(Aᵀ) = det(A)
- det(A⁻¹) = 1/det(A) if det(A) ≠ 0
- Swapping two rows changes sign
- Multiplying row by k multiplies det by k
About Matrix Determinant Calculator
The Matrix Determinant Calculator computes the determinant of a square matrix. The determinant is a scalar value that indicates whether a matrix is invertible and encodes geometric properties of linear transformations.
When to Use This Calculator
- Invertibility: Check if matrix has inverse (det ≠ 0)
- Linear Systems: Determine if Ax = b has unique solution
- Volume Scaling: Find volume change under transformation
- Cramer's Rule: Solve systems using determinants
- Eigenvalues: Find eigenvalues from characteristic polynomial
Why Use Our Calculator?
- ✅ Multiple Sizes: Supports 2×2, 3×3, and 4×4 matrices
- ✅ Laplace Expansion: Uses cofactor expansion method
- ✅ Clear Result: Shows determinant and singularity status
- ✅ Educational: Helps understand determinants
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Singular: det(A) = 0, matrix has no inverse
- Non-Singular: det(A) ≠ 0, matrix is invertible
- Volume: |det(A)| is volume scaling factor
- Orientation: Sign indicates orientation preservation
- Rank: For square matrix, rank = n iff det ≠ 0
Applications
System Solving: Ax = b has unique solution iff det(A) ≠ 0.
Transformations: Determinant tells how much area/volume scales.
Frequently Asked Questions
What does a determinant of zero mean?
A determinant of zero indicates the matrix is singular (non-invertible). This means the transformation collapses space, rows/columns are linearly dependent, and the matrix has no inverse.
Can a non-square matrix have a determinant?
No, determinants are only defined for square matrices. For rectangular matrices, concepts like pseudo-determinant or determinant of Gramian matrix are used.
Is the determinant always positive?
No! The determinant can be negative. The sign indicates orientation: positive preserves orientation, negative reverses it. The absolute value |det| tells the scaling factor.
How is determinant related to eigenvalues?
The determinant equals the product of all eigenvalues: det(A) = λ₁ × λ₂ × ... × λₙ. So det = 0 iff at least one eigenvalue is zero.
Can I find determinant of 5×5 or larger matrices?
Yes, but the calculator currently supports up to 4×4. For larger matrices, use row operations to reduce, or compute det(A) = det(L) × det(U) from LU decomposition.