Quick answer
The determinant is computed by cofactor expansion along the first row. For [[1, 2], [3, 4]] it is 1 × 4 − 2 × 3 = −2.
Enter a square matrix with rows separated by semicolons. The calculator returns the determinant.
Related tools: Matrix Calculator: Determinant, trace, transpose and 2×2 inverse of a matrix. Matrix Multiplication Calculator: Multiply two matrices. Slope Calculator: Slope, angle, distance and line equation through two points.
How to use the Matrix Determinant Calculator
- Type matrix (rows separated by ;), for example 1 2; 3 4.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Matrix (rows separated by ;) | Text, such as 1 2; 3 4 | — |
Formula and method
Formula
Laplace (cofactor) expansion along the first row
IEEE-754 double precision; shown to at most 6 decimal places.
Recursive cofactor expansion; the matrix must be square.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Matrix (rows separated by ;) = 1 2; 3 4.
| Result | Value |
|---|---|
| Determinant | -2 |
Example 2
Inputs: Matrix (rows separated by ;) = 1 2 3; 0 1 4; 5 6 0.
| Result | Value |
|---|---|
| Determinant | 1 |
Example 3
Inputs: Matrix (rows separated by ;) = 7.
| Result | Value |
|---|---|
| Determinant | 7 |
The first line is the headline result (determinant). IEEE-754 double precision; shown to at most 6 decimal places.
3×3 example
For [[2, 0, 1], [1, 3, 2], [1, 1, 1]]: 2(3 − 2) − 0(1 − 2) + 1(1 − 3) = 2 − 0 − 2 = 0, so that matrix is singular.
What it tells you
A non-zero determinant means the matrix is invertible and the linear system has a unique solution. The absolute value is the factor by which the matrix scales areas or volumes. Only square matrices have determinants.
Reading the value
For a 2×2 matrix with rows (a b) and (c d) the determinant is ad − bc, so 1·4 − 2·3 = −2. The size of the determinant is the factor by which the matrix scales areas, and the sign shows whether orientation flips.
A determinant of 0 means the matrix is singular: its rows are dependent, such as (1 2) and (2 4), and it has no inverse. Large matrices computed in floating point may show a tiny nonzero value instead of an exact zero.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Matrix (rows separated by ;): entering "1 2 3; 4 5 6" shows "The matrix must be square." beside that field and no result.
Choosing between the Matrix Determinant Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Matrix Calculator](/math/matrix-calculator/) | matrix | Determinant |
| [Matrix Multiplication Calculator](/math/matrix-multiplication-calculator/) | matrix A and matrix B | A × B |
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
| [LCM Calculator](/math/lcm-calculator/) | numbers | Least common multiple |
| [Factor Calculator](/math/factor-calculator/) | number | Factors |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- Square matrices only
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for matrix (rows separated by ;) to confirm this.
Frequently asked questions
What is the determinant of the identity?
The determinant of the identity matrix is 1.
Does swapping rows change it?
Swapping two rows multiplies the determinant by −1.
Is it zero for singular matrices?
Yes. A singular matrix has determinant 0 and no inverse.
What does a negative determinant mean?
It means the transformation reverses orientation, like a reflection, while its size still gives the area or volume scale factor.
Can the determinant tell me whether the inverse exists?
Yes. A nonzero determinant means the matrix is invertible and a zero determinant means it is not.
What does the Matrix Determinant Calculator calculate?
Determinant of a square matrix (up to 6×6). It returns Determinant.
What formula does the Matrix Determinant Calculator use?
Laplace (cofactor) expansion along the first row. Recursive cofactor expansion; the matrix must be square.
What do I need to enter in the matrix Determinant Calculator?
You enter Matrix (rows separated by ;).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Square matrices only.
What is an example of a matrix Determinant result?
With matrix (rows separated by ;) 1 2; 3 4, the determinant is -2.
Where does the method come from?
The method follows: OpenStax, Algebra and Trigonometry 2e (Rice University): matrices, determinants and systems of equations.
Related calculators
- Matrix Calculator — Determinant, trace, transpose and 2×2 inverse of a matrix.
- Matrix Multiplication Calculator — Multiply two matrices.
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
- Factor Calculator — List every factor of a positive whole number.
- Quadratic Formula Calculator — Solve ax² + bx + c = 0, including complex roots.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): matrices, determinants and systems of equations
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.