Cofactor Expansion (Determinant) Calculator
Compute the determinant of a 2×2 or 3×3 matrix via cofactor expansion (Laplace expansion). See minors, cofactors, and a full step-by-step expansion breakdown.
Cofactor Matrix & Adjugate Calculator
Compute the cofactor matrix, adjugate (classical adjoint), determinant, and inverse of a 2×2 or 3×3 matrix with step-by-step cofactor breakdowns and visual grids.
Matrix A
Determinant⧉
1.00
det(A) = 1.00
Is Invertible?⧉
Yes
det ≠ 0, inverse exists
Cofactor Matrix⧉
3×3 matrix
[-24, 20, -5]
[18, -15, 4]
[5, -4, 1]
Adjugate (adj A)⧉
3×3 matrix
[-24, 18, 5]
[20, -15, -4]
[-5, 4, 1]
Inverse (A⁻¹)⧉
3×3 matrix
[-24, 18, 5]
[20, -15, -4]
[-5, 4, 1]
1 / det(A)⧉
1.000000
Scalar multiplier for A⁻¹ = (1/det) × adj(A)
Cofactor Matrix Grid
| Col 1 | Col 2 | Col 3 | |
|---|---|---|---|
| Row 1 | -24.00 | 20.00 | -5.00 |
| Row 2 | 18.00 | -15.00 | 4.00 |
| Row 3 | 5.00 | -4.00 | 1.00 |
Adjugate Matrix Grid
| Col 1 | Col 2 | Col 3 | |
|---|---|---|---|
| Row 1 | -24.00 | 18.00 | 5.00 |
| Row 2 | 20.00 | -15.00 | -4.00 |
| Row 3 | -5.00 | 4.00 | 1.00 |
Inverse Matrix (A⁻¹)
| Col 1 | Col 2 | Col 3 | |
|---|---|---|---|
| Row 1 | -24 | 18 | 5 |
| Row 2 | 20 | -15 | -4 |
| Row 3 | -5 | 4 | 1 |
Cofactor Breakdown
| Position | Minor Mᵢⱼ | Sign (−1)^(i+j) | Cofactor Cᵢⱼ |
|---|---|---|---|
| (1,1) | -24.00 | + | -24.00 |
| (1,2) | -20.00 | − | 20.00 |
| (1,3) | -5.00 | + | -5.00 |
| (2,1) | -18.00 | − | 18.00 |
| (2,2) | -15.00 | + | -15.00 |
| (2,3) | -4.00 | − | 4.00 |
| (3,1) | 5.00 | + | 5.00 |
| (3,2) | 4.00 | − | -4.00 |
| (3,3) | 1.00 | + | 1.00 |
Cofactor Magnitudes — Visual
C(1,1)
-24.00
C(1,2)
20.00
C(1,3)
-5.00
C(2,1)
18.00
C(2,2)
-15.00
C(2,3)
4.00
C(3,1)
5.00
C(3,2)
-4.00
C(3,3)
1.00
■ Positive ■ Negative
Planning notes, formulas, and examples
About the Cofactor Matrix & Adjugate Calculator
The cofactor matrix of a square matrix A is formed by replacing each element aᵢⱼ with its cofactor Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ, where Mᵢⱼ is the minor (the determinant of the sub-matrix obtained by deleting row i and column j). The adjugate (classical adjoint) is the transpose of the cofactor matrix.
The adjugate is essential for computing the matrix inverse using the formula A⁻¹ = (1/det(A)) × adj(A). This formula is elegant for theoretical work and practical for 2×2 and 3×3 matrices. It also appears in Cramer's rule for solving systems of linear equations.
This calculator lets you enter a 2×2 or 3×3 matrix and see its cofactor matrix, adjugate, determinant, and inverse (when it exists). Each cofactor is displayed in a grid that shows the sign pattern, minor value, and final cofactor value. The adjugate is presented alongside the original matrix for easy comparison. Bar charts visualize the relative magnitudes of cofactors across the matrix, making patterns immediately apparent.
When This Page Helps
Cofactor Matrix & Adjugate Calculator helps you solve cofactor matrix & adjugate problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter your inputs once and immediately inspect Determinant, Is Invertible?, Cofactor Matrix to validate your work.
How to Use the Inputs
- Select the mode, method, or precision options that match your cofactor matrix & adjugate problem.
- Read Determinant first, then use Is Invertible? to confirm your setup is correct.
- Try a preset such as "2 × 2" to test a known case quickly.
- Compare the result with the formula and worked example so you can catch input, rounding, or setup mistakes.
Formula used
Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ; adj(A) = [Cᵢⱼ]ᵀ; A⁻¹ = (1/det A) × adj(A)Example Calculation
Result: Determinant shown by the calculator
Using the preset "2 × 2", the calculator evaluates the cofactor matrix & adjugate setup, applies the selected algebra rules, and reports Determinant with supporting checks so you can verify each transformation.
Tips & Best Practices
- The adjugate is the TRANSPOSE of the cofactor matrix — don't confuse the two.
- If det(A) = 0, the inverse does not exist.
- For a 2×2 matrix, the adjugate has a simple pattern: swap diagonal, negate off-diagonal.
- The formula A × adj(A) = det(A) × I is always true, even when det = 0.
- Cofactors follow a checkerboard sign pattern: +−+, −+−, +−+.
How This Cofactor Matrix & Adjugate Calculator Works
This calculator takes the problem inputs and applies the relevant cofactor matrix & adjugate relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Interpreting Results
Start with the primary output, then use Determinant, Is Invertible?, Cofactor Matrix, Adjugate (adj A) to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
Study Strategy
Sources & Methodology
Last updated:
Frequently Asked Questions
The cofactor matrix replaces each element aᵢⱼ with its cofactor Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ. It captures each element's contribution to the determinant.
The adjugate is the transpose of the cofactor matrix. It is used in the formula A⁻¹ = (1/det) × adj(A) to compute the matrix inverse.
In classical linear algebra, "adjoint" or "classical adjoint" means the adjugate (transpose of cofactors). In functional analysis, "adjoint" can mean the conjugate transpose. This calculator uses the classical definition.
The inverse does not exist when det(A) = 0 (the matrix is singular). The adjugate still exists, but dividing by zero is undefined.
Multiply A × A⁻¹ — if you get the identity matrix, the inverse is correct. This calculator computes it as (1/det) × adj(A).
Yes, cofactors generalize to any square matrix. However, for n > 4, this approach is impractical. Use Gaussian elimination or LU decomposition instead.
More in this topic
Characteristic Polynomial Calculator
Find the characteristic polynomial, eigenvalues, trace, and determinant of a 2×2 or 3×3 matrix with step-by-step computation and eigenvalue visualization.
Matrix Inverse Calculator
Find the inverse of a matrix up to 5×5 using the cofactor/adjugate method, with step-by-step cofactor table, verification A×A⁻¹=I, condition number analysis, and singular detection.