BYTETOOLS

Determinant Calculator

Compute the determinant of any square matrix up to 6×6 instantly. See the full cofactor expansion for 2×2 and 3×3 matrices, worked step by step.

49
Determinant (3×3)
Invertible
Matrix type

Cofactor expansion

Cofactor expansion along the first row:
det = 2 × |0 -1; 4 5| − -3 × |2 -1; 1 5| + 1 × |2 0; 1 4|
det = 2 × 4 − -3 × 11 + 1 × 8
det = 8 − -33 + 8 = 49

What is the Determinant Calculator?

The ByteTools Determinant Calculator computes the determinant of any square matrix from 2×2 up to 6×6.

  • Determinants for square matrices up to 6×6
  • Numerically stable LU decomposition with partial pivoting
  • Full written-out cofactor expansion for 2×2 and 3×3
  • Flags singular matrices (determinant = 0)
  • Accepts decimals and negative entries
  • 100% private — computed entirely in your browser

How to use the Determinant Calculator

  1. 1

    Type the square matrix: one row per line, numbers separated by spaces or commas.

  2. 2

    Check the detected size — the matrix must have as many rows as columns.

  3. 3

    Read the determinant instantly.

  4. 4

    For 2×2 and 3×3 inputs, follow the printed cofactor expansion steps.

  5. 5

    Copy the result with one click.

About the Determinant Calculator

The ByteTools Determinant Calculator computes the determinant of any square matrix from 2×2 up to 6×6. Type the matrix as rows of numbers and the determinant appears instantly, computed by LU decomposition with partial pivoting — a numerically stable method that also works well for larger matrices.

For 2×2 and 3×3 matrices the tool additionally prints the classic cofactor expansion with every product written out, exactly the way it is taught in class, so you can follow the arithmetic line by line. A determinant of zero is flagged clearly, since it means the matrix is singular and has no inverse.

The calculation runs 100% locally in your browser using JavaScript. Nothing you type is uploaded or stored, results are copyable in one click, and the tool works offline.

Frequently asked questions

How do you find the determinant of a 3x3 matrix?

Expand along the first row: multiply each entry by the determinant of the 2×2 matrix left after deleting its row and column, alternating signs +, −, +. For [[a,b,c],[d,e,f],[g,h,i]] that gives a(ei − fh) − b(di − fg) + c(dh − eg). This tool prints that exact expansion.

What does a determinant of zero mean?

It means the matrix is singular: its rows or columns are linearly dependent, it has no inverse, and a linear system with that coefficient matrix has either no solution or infinitely many. Geometrically, the matrix squashes space onto a lower dimension.

Can a determinant be negative?

Yes. The sign carries information: a negative determinant means the transformation flips orientation, like a reflection. The absolute value tells you how areas (2×2) or volumes (3×3) are scaled by the matrix.

What method does this calculator use for larger matrices?

It uses LU decomposition with partial pivoting: the matrix is reduced to upper-triangular form by row elimination, and the determinant is the product of the diagonal entries, with the sign adjusted for each row swap. This is far faster and more accurate than cofactor expansion for 4×4 and above.

Does the determinant exist for non-square matrices?

No. Determinants are only defined for square matrices — the same number of rows and columns. If you enter a rectangular matrix, the calculator tells you the shape it found and asks for a square one.

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