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3×3 Determinant Calculator

Find a 3×3 matrix determinant.

How it is calculated

The matrix is read row by row as [a b c; d e f; g h i] and expanded along the first row: det = a(ei − fh) − b(di − fg) + c(dh − eg). Each bracket is the 2×2 determinant left after crossing out the row and column of that first-row entry, and the signs alternate +, −, +. All nine cells must be filled in; decimals and negatives are fine. The result is rounded to two decimals. A determinant of 0 means the matrix is singular: it has no inverse and its rows are linearly dependent.

Worked example

Matrix rows: (2, 1, 3), (0, 4, 1), (5, 2, 1). First term: 2·(4·1 − 1·2) = 2·2 = 4. Second term: −1·(0·1 − 1·5) = −1·(−5) = 5. Third term: 3·(0·2 − 4·5) = 3·(−20) = −60. Sum: 4 + 5 − 60 = -51.00. The matrix is non-singular because the determinant is not zero.

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Common mistakes

  • Forgetting the minus sign in front of the middle term, −b(di − fg).
  • Entering the matrix column by column instead of row by row. Transposing does not change a determinant, but a mixed-up order does.
  • Calculating the 2×2 minors as ei − hf with the wrong cells, instead of ei − fh using the cells that remain.
  • Reading a small determinant as meaning the entries are small; scaling every entry by k multiplies the determinant by k³.
  • Assuming det = 0 means an error; it is a valid result that signals a singular matrix.

About this calculator

This calculator finds the determinant of a 3×3 matrix using cofactor expansion (equivalent to the rule of Sarrus), combining products of the entries along diagonals to produce a single scalar value.

The determinant tells you whether a matrix can be inverted — a determinant of zero means it can't — and its magnitude represents how much a linear transformation scales area or volume. Engineering and computer graphics students use it to check system solvability, and it's a routine step in linear algebra coursework and 3D transformation math.

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Comments (1)

  • Editorial team

    A determinant of exactly zero means the three rows are linearly dependent, so the matrix has no inverse and a 3-equation system built from it has no single unique solution. Row order matters: swapping any two rows flips the sign, which makes a good self-check. Because inputs are plain decimals, a tiny nonzero result can be rounding, not a real difference. Which application brings you here, equations, area or volume? Add a line about your matrix.

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