Determinant Calculator

Calculate the determinant of a square matrix

Matrix Determinant Calculator – Compute Det of Any Matrix

Calculate the determinant of any square matrix up to 10×10 with our free online determinant calculator. Enter values via textarea or row-by-row input with cofactor expansion steps shown.

Matrix Size:

Enter matrix values (each row on a new line, values separated by spaces or commas)

Current matrix: 3×3 | Detected from your input

What is a Determinant?

The determinant is a scalar value that can be computed from a square matrix. It provides important information about the matrix and is used in many areas of mathematics including solving systems of linear equations, finding matrix inverses, and calculating areas and volumes.

A determinant of 0 means the matrix is singular (not invertible). A non-zero determinant means the matrix has an inverse.

Determinant Formulas

2×2 Matrix

|a b|
|c d| = ad - bc

Multiply diagonals and subtract

3×3 Matrix (Cofactor Expansion)

det = a(ei - fh) - b(di - fg) + c(dh - eg)

Expand along first row using minors

Properties of Determinants

Key Properties

• det(AB) = det(A) × det(B)
• det(A⁻¹) = 1/det(A)
• det(Aᵀ) = det(A)
• det(kA) = kⁿ × det(A) for n×n matrix

When det = 0

• Matrix is not invertible
• Rows/columns are linearly dependent
• System has no unique solution
• Transformation collapses space

Frequently Asked Questions

What does a determinant of 0 mean?

A determinant of 0 means the matrix is singular – it has no inverse. The rows or columns are linearly dependent, and the matrix transformation collapses space to a lower dimension.

Can determinants be negative?

Yes, determinants can be positive, negative, or zero. A negative determinant indicates the matrix includes a reflection (it reverses orientation).

How is the determinant used?

Determinants are used to check if a matrix is invertible, solve systems of equations (Cramer's rule), find eigenvalues, calculate volumes in geometry, and in calculus for change of variables.

What is cofactor expansion?

Cofactor expansion is a method to calculate determinants by expanding along a row or column. Each element is multiplied by its cofactor (signed minor determinant) and summed.

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