# Determinant Calculator

Calculate the determinant of a 2x2 or 3x3 matrix with step-by-step cofactor expansion. Shows whether the matrix is invertible.

## What this calculates

Find the determinant of a 2x2 or 3x3 matrix with step-by-step work. The determinant tells you whether a matrix is invertible and plays a key role in solving systems of linear equations, computing eigenvalues, and more.

## Inputs

- **Matrix Size** — options: 2 x 2, 3 x 3 — Choose the dimension of your square matrix.
- **a₁₁ (Row 1, Col 1)**
- **a₁₂ (Row 1, Col 2)**
- **a₁₃ (Row 1, Col 3)** — Only used for 3x3 matrices.
- **a₂₁ (Row 2, Col 1)**
- **a₂₂ (Row 2, Col 2)**
- **a₂₃ (Row 2, Col 3)** — Only used for 3x3 matrices.
- **a₃₁ (Row 3, Col 1)** — Only used for 3x3 matrices.
- **a₃₂ (Row 3, Col 2)** — Only used for 3x3 matrices.
- **a₃₃ (Row 3, Col 3)** — Only used for 3x3 matrices.

## Outputs

- **Determinant** — The determinant of the matrix.
- **Invertible?** — formatted as text — A matrix is invertible if and only if its determinant is non-zero.
- **Calculation Steps** — formatted as text — Step-by-step expansion of the determinant.

## Details

**2x2 Determinant**

For a 2x2 matrix [[a, b], [c, d]], the determinant is:

**det = ad - bc**

Example: [[3, 7], [1, 5]] has det = (3)(5) - (7)(1) = 15 - 7 = 8.

**3x3 Determinant (Cofactor Expansion)**

For a 3x3 matrix, expand along the first row:

det = a11(a22 a33 - a23 a32) - a12(a21 a33 - a23 a31) + a13(a21 a32 - a22 a31)

Each 2x2 block in parentheses is called a minor. The alternating +/- signs come from the cofactor sign pattern.

**What the Determinant Tells You**

- **Non-zero determinant:** The matrix is invertible (has a unique inverse). A system of equations with this coefficient matrix has exactly one solution.
- **Zero determinant:** The matrix is singular (not invertible). The corresponding system has either no solution or infinitely many solutions.
- The absolute value of the determinant also represents the scale factor by which the matrix transforms areas (2D) or volumes (3D).

## Frequently Asked Questions

**Q: What does the determinant of a matrix mean?**

A: The determinant is a single number that encodes important properties of a square matrix. If it is non-zero, the matrix is invertible and the associated system of equations has a unique solution. The absolute value of the determinant also represents how the matrix scales areas (2D) or volumes (3D).

**Q: How do you calculate a 2x2 determinant?**

A: For a 2x2 matrix [[a, b], [c, d]], multiply the diagonals and subtract: det = ad - bc. For example, the matrix [[4, 3], [2, 1]] has determinant (4)(1) - (3)(2) = 4 - 6 = -2.

**Q: What does a zero determinant mean?**

A: A zero determinant means the matrix is singular, meaning it has no inverse. Geometrically, it collapses space into a lower dimension (e.g., a plane into a line). For a system of equations, it means there is no unique solution.

**Q: Can determinants be negative?**

A: Yes, determinants can be any real number: positive, negative, or zero. A negative determinant means the transformation reverses orientation (like a mirror reflection). The sign does not affect invertibility -- only whether the determinant is zero matters.

---

Source: https://vastcalc.com/calculators/math/determinant
Category: Math
Last updated: 2026-04-08