Determinant Calculator

The determinant is a single number calculated from a square matrix. It tells you whether the matrix has an inverse and how much it scales areas or volumes. This calculator uses ad − bc for 2 × 2 matrices, cofactor expansion for 3 × 3, and row reduction for larger ones.

Separate entries with spaces and rows with new lines or semicolons.
Try:

How to use the determinant calculator

  1. Type the square matrix, one row per line.
  2. Separate entries with spaces or commas.
  3. Press Calculate to see the determinant and the expansion steps.

Formula

det⁡(abcd)=ad−bc\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc

Worked example: 2×2

  • Square matrix: 4 7 2 6

✓ Answer checked

∣A∣=10|A| = 10
Invertible?
Yes — the determinant is not 0.
−551015−2246810
The matrix maps the unit square (green, area 1) to a parallelogram (blue) whose area is |det| = 10.
Step-by-step working (1 step)

  1. Use the 2×2 formula 2×2 determinant

    Multiply the main diagonal and subtract the product of the other diagonal.

    ∣abcd∣=ad−bc=(4)(6)−(7)(2)=10\begin{gathered}\begin{vmatrix}a&b\\c&d\end{vmatrix} = ad - bc = (4)(6) - (7)(2) = 10\end{gathered}
Formulas used
det⁡(abcd)=ad−bc\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc
det⁡A=∑j(−1)1+ja1jM1j\det A = \sum_j (-1)^{1+j} a_{1j} M_{1j}
How this was checked
  • ✓ Row reduction gives the same determinant (10).

Frequently asked questions

How do I find a 2×2 determinant?

Multiply the main diagonal and subtract the product of the other diagonal: ad − bc.

What does a determinant of 0 mean?

The matrix is singular: it has no inverse, and it squashes space into a lower dimension.

What does the determinant mean geometrically?

For a 2×2 matrix, its absolute value is the area of the parallelogram the unit square is mapped to.

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