The determinant of a 2×2 matrix [[a,b],[c,d]] is ad − bc. For [[4,7],[2,6]] it is 4×6 − 7×2 = 10. A square matrix is invertible exactly when its determinant is not zero, and for 3×3 and 4×4 matrices the calculator uses elimination to get the value.
How do you calculate a determinant?
For a 2×2 matrix the rule is simple: multiply the main diagonal and subtract the product of the other diagonal. For a 3×3 matrix you can expand along the first row: a(ei − fh) − b(di − fg) + c(dh − eg), where the matrix is [[a,b,c],[d,e,f],[g,h,i]]. Larger matrices use cofactor expansion or, much faster, Gaussian elimination: reduce the matrix to upper-triangular form and multiply the diagonal entries, flipping the sign for each row swap. That is how this calculator works, so it handles 4×4 as easily as 2×2.
What does the determinant tell you?
- Invertibility: a non-zero determinant means the matrix has an inverse, and zero means it is singular.
- Volume scaling: the absolute value is the factor by which the matrix scales area (2D) or volume (3D). A determinant of 3 triples areas.
- Orientation: a negative determinant means the transformation flips orientation, like a mirror.
- Systems of equations: A·x = b has a unique solution when det(A) ≠ 0 (Cramer’s rule).
Worked example (3×3)
Take [[2,1,1],[1,3,2],[1,0,0]]. Expanding along the bottom row, which has two zeros, gives 1 × (1×2 − 1×3) = −1. The calculator confirms det = −1, so the matrix is invertible, and its rank is 3.
Useful properties
| Property | Rule |
|---|---|
| Product | det(AB) = det(A) × det(B) |
| Transpose | det(Aᵀ) = det(A) |
| Inverse | det(A⁻¹) = 1 ÷ det(A) |
| Row swap | Changes the sign |
| Row multiplied by k | Determinant multiplied by k |
| Two equal rows | Determinant is 0 |
Next steps: compute the matrix inverse in the inverse matrix calculator, or explore other operations in the main matrix calculator. The determinant is also the product of the eigenvalues, which you can verify in the eigenvalue calculator.
Determinants and systems of equations
The determinant decides whether a system of linear equations has exactly one solution. Suppose 2x + y = 5 and x + 3y = 10. The coefficient matrix [[2,1],[1,3]] has determinant 2×3 − 1×1 = 5, which is not zero, so there is one solution. Cramer’s rule gives it directly: x = det[[5,1],[10,3]] ÷ 5 = (15 − 10) ÷ 5 = 1 and y = det[[2,5],[1,10]] ÷ 5 = (20 − 5) ÷ 5 = 3. If the determinant had been zero, the equations would describe parallel or identical lines, giving no solution or infinitely many.
The same idea scales up: a 3×3 determinant tells you whether three planes meet in a single point, and the volume of the parallelepiped spanned by three vectors is the absolute value of the determinant of the matrix formed by them.
How to use this calculator
- Choose the matrix size (2×2, 3×3 or 4×4) and the operation.
- Type your numbers into the matrix grids. Decimals and negatives are fine.
- The result updates instantly, including checks such as A × A⁻¹.
- Change the operation to explore the same matrix in other ways.
Key figures at a glance
| 2×2 | ad − bc |
|---|---|
| Singular if | det = 0 |
| Method here | Gaussian elimination |
| Product of | the eigenvalues |
Frequently asked questions
Expand along a row: a(ei − fh) − b(di − fg) + c(dh − eg) for the matrix [[a,b,c],[d,e,f],[g,h,i]].
The matrix is singular: it has no inverse, its rows or columns are linearly dependent and its rank is below full.
Yes. A negative determinant means the transformation reverses orientation.
1.
Yes, the determinant equals the product of all eigenvalues, counted with multiplicity.