Matrix Determinant Calculator

Calculate the determinant of any square matrix with detailed step-by-step solutions.

Matrix Input
Supports 2×2 up to 10×10 square matrices
One row per line, space or comma separated
Quick Presets
Result
Enter a square matrix and click "Calculate Determinant" to see the result.
Step-by-Step Solution
Detailed steps will appear here after calculation.
About Determinants

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, including whether it's invertible.

Properties:

  • det(I) = 1 (identity matrix)
  • det(AB) = det(A) × det(B)
  • det(AT) = det(A)
  • det(kA) = kn × det(A) for n×n matrix
  • If det(A) = 0, the matrix is singular (not invertible)

Methods:

  • LU Decomposition: Fastest for large matrices, O(n³) complexity
  • Cofactor Expansion: Educational, shows formula clearly, O(n!) complexity
  • Gaussian Elimination: Row reduction method, O(n³) complexity
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Matrix Determinant: FAQ

How do I calculate the determinant of a matrix?

Enter a square matrix and click Calculate. The tool shows step‑by‑step methods such as cofactor expansion and row operations; for larger sizes it may use LU decomposition for efficiency.

What sizes and methods are supported?

This calculator supports square matrices from 2×2 up to 10×10 and can display cofactor expansion steps, row‑operation reductions, and LU‑based computations.

What does det(A) = 0 mean?

det(A) = 0 indicates the matrix is singular: rows/columns are linearly dependent, rank is less than n, and A is not invertible.

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