3x3 Determinant Calculator

Compute the determinant of any 3×3 matrix in one click. Determinants reveal whether a matrix is invertible, the signed volume scaling of a linear transformation, and the area or volume of parallelograms generated by vectors.

Row 1, column 1 entry
Row 1, column 2 entry
Row 1, column 3 entry
Row 2, column 1 entry
Row 2, column 2 entry
Row 2, column 3 entry
Row 3, column 1 entry
Row 3, column 2 entry
Row 3, column 3 entry

Double-check results when using determinants in engineering or physics analyses—rounding input values can change the outcome.

Examples

  • Matrix [[3, −1, 0.8], [4, 5.5, 2], [7, −3, 6.1]] ⇒ determinant 136.72, so the matrix is invertible and scales volume by 136.72.
  • Matrix [[2, 0, 1], [3, 0, 0], [5, 1, 1]] ⇒ determinant 3, matching the cofactor expansion by hand.
  • Matrix [[6, 1, 1], [4, −2, 5], [2, 8, 7]] ⇒ determinant −306, indicating a reflection combined with a 306× volume scaling.

FAQ

Can I input decimals or fractions?

Yes. Enter any real numbers—including decimals or negative values. For fractions, convert them to decimal form (e.g., 1/3 becomes 0.3333).

What does a negative determinant tell me?

A negative value indicates that the associated linear transformation flips orientation, such as converting a right-handed coordinate system to a left-handed one, while scaling volume by the absolute value of the determinant.

How is the formula derived?

The expression a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁) comes from expanding along the first row using Laplace's cofactor expansion.

Do units matter for determinants?

If your matrix entries represent scaled coordinates, the determinant carries the combined unit raised to the matrix dimension. In most algebra problems it's treated as dimensionless, but in physics it can correspond to area or volume scaling.

Additional Information

  • A zero determinant means the matrix is singular and cannot be inverted; the rows or columns are linearly dependent.
  • Positive determinants preserve orientation, while negative determinants indicate an orientation flip in the transformation.
  • Use the determinant when solving 3×3 systems via Cramer's rule, computing eigenvalues, or finding the Jacobian in multivariable calculus.
  • For larger matrices, LU decomposition or row-reduction algorithms are more efficient than cofactor expansion.

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