Matrix Determinant
2×2 and 3×3 determinant
About This Calculator
The determinant is a scalar value computed from a square matrix that encodes key geometric and algebraic properties. A non-zero determinant means the matrix is invertible; a zero determinant means the system is singular (no unique solution). Determinants measure the scaling factor of the linear transformation and relate to area (2×2) or volume (3×3) changes.
Formula
2×2: det [[a,b],[c,d]] = ad − bc
3×3: Laplace expansion along first row: a(ei−fh) − b(di−fg) + c(dh−eg)
Properties: det(AB)=det(A)×det(B); det(Aᵀ)=det(A); det(λA)=λⁿdet(A) for n×n matrix
Example Calculation
Find det of [[3,8],[4,6]] and [[2,1,0],[3,−1,2],[1,4,−3]].
- 2×2: det = 3×6 − 8×4 = 18 − 32 = −14
- 3×3 row 1 expansion: 2(3−8)−1(−9−2)+0 = 2(−5)−1(−11) = −10+11 = 1
2×2 det = −14; 3×3 det = 1
Determinant Properties
| Property | Meaning |
|---|---|
| det ≠ 0 | Matrix is invertible (non-singular) |
| det = 0 | Matrix is singular — rows/cols are linearly dependent |
| det = 1 | Area/volume preserving transformation |
| det < 0 | Transformation includes a reflection |
| det = ad−bc for 2×2 | Area of parallelogram formed by rows |
Frequently Asked Questions
What does a determinant of zero mean?
A zero determinant means the matrix is singular: its rows (or columns) are linearly dependent, so the corresponding system of equations has either no solution or infinitely many solutions. The matrix cannot be inverted.
How is the determinant related to area?
For a 2×2 matrix with row vectors a and b, |det| equals the area of the parallelogram they form. For a 3×3 matrix, |det| equals the volume of the parallelepiped. A negative determinant indicates a reflection.
What is the cofactor expansion method?
Cofactor expansion (Laplace expansion) computes the determinant by expanding along any row or column, multiplying each element by its cofactor (signed minor). Choose the row/column with the most zeros to minimize calculations.
What is the difference between a minor and a cofactor?
The minor Mᵢⱼ of element aᵢⱼ is the determinant of the submatrix obtained by deleting row i and column j. The cofactor Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ — it includes the checkerboard sign pattern.