Determinant Calculator

Calculate the determinant of 2×2 and 3×3 matrices. The determinant determines if a matrix is invertible, finds area/volume scaling factors, and solves systems via Cramer's rule.

∑ Linear Algebra📐 det(A) = ad - bc (2×2)🔢 Math
Matrix size
a₁₁
a₁₂
a₂₁
a₂₂
a₁₃ (3×3 only)
a₂₃
a₃₁
a₃₂
a₃₃
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Determinant Calculator—det(A) = ad - bc (2×2)scalar

Step-by-Step Examples

Example 1
2×2

[[3,8],[4,6]].

  • det = 3×6 − 8×4 = 18 − 32 = −14
  • Invertible: det ≠ 0
✓ det = −14
Example 2
Identity Matrix

[[1,0],[0,1]].

  • det = 1×1 − 0×0 = 1
  • Identity always has det = 1
✓ det = 1
Example 3
3×3

[[1,2,3],[4,5,6],[7,8,9]].

  • det = 1(45−48) − 2(36−42) + 3(32−35)
  • = 1(−3) − 2(−6) + 3(−3) = −3+12−9 = 0
  • Singular: rows are linearly dependent
✓ det = 0 (singular)

Real-World Applications

📊
Linear Algebra
Core operation for solving systems (Cramer's rule) and finding matrix inverses.
📐
Geometry
|det(A)| = area scaling factor of transformation (2×2) or volume (3×3).
🧪
Physics
Jacobian determinant used in change of variables for integration.
💻
Graphics
Determinant sign tells if transformation flips orientation.

Common Mistakes to Avoid

⚠️
Forgetting the minus sign on middle term (3×3)

Cofactor expansion: +a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃. The middle cofactor is negative.

⚠️
Swapping rows changes sign

Each row swap multiplies the determinant by −1. Two swaps return to original sign.

⚠️
det(AB) = det(A)×det(B)

Determinant is multiplicative. Very useful property, but det(A+B) ≠ det(A)+det(B).

Frequently Asked Questions

What does the determinant measure? ▾
The scaling factor of the area (2D) or volume (3D) when the matrix transformation is applied. det=2 means the transformation doubles areas.
When is a matrix invertible? ▾
If and only if its determinant is non-zero. det=0 means the matrix is singular and has no inverse.
What is Cramer's rule? ▾
Solves Ax=b using determinants: x_i = det(A_i)/det(A) where A_i replaces the i-th column with b.
What is a cofactor? ▾
The cofactor C_ij = (−1)^(i+j) × det(minor M_ij). Used in cofactor expansion and finding matrix inverse.
What is an eigenvalue? ▾
Scalars λ where Av = λv for nonzero vector v. The characteristic polynomial det(A−λI)=0 gives eigenvalues.

Related Math Calculators

∑ Matrix∑ Systems of Equations∑ Vector∑ Linear Equation∑ Complex Numbers

Formula Explorer connections

Interpretation: This relationship represents vectors, linear systems or transformations using matrix structure and vector operations. Assumption: Matrix dimensions must be compatible. Inversion, decomposition and unique solutions require rank, conditioning or nonzero-determinant conditions.

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