Quick answer
For a square matrix the calculator gives the determinant, trace, transpose and inverse. For [[1, 2], [3, 4]]: determinant −2, trace 5, transpose [[1, 3], [2, 4]] and inverse [[−2, 1], [1.5, −0.5]].
Enter a matrix with rows separated by semicolons. The calculator returns its determinant, trace, transpose and inverse.
Related tools: Matrix Determinant Calculator: Determinant of a square matrix (up to 6×6). Matrix Multiplication Calculator: Multiply two matrices. Slope Calculator: Slope, angle, distance and line equation through two points.
How to use the Matrix 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
det, trace, transpose; inverse (2×2) = (1/det)[[d, −b], [−c, a]]
IEEE-754 double precision; shown to at most 6 decimal places.
Inverse is provided for 2×2 matrices with non-zero determinant.
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 |
| Trace | 5 |
| Transpose | 1 3; 2 4 |
| Inverse | -2 1; 1.5 -0.5 |
Example 2
Inputs: Matrix (rows separated by ;) = 1 2 3; 4 5 6.
| Result | Value |
|---|---|
| Determinant | n/a (not square) |
| Transpose | 1 4; 2 5; 3 6 |
The first line is the headline result (determinant). Below it the matrix Calculator lists trace, transpose and inverse, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
2×2 formulas
det = ad − bc = 1 × 4 − 2 × 3 = −2. Inverse = (1 ÷ det) × [[d, −b], [−c, a]] = −0.5 × [[4, −2], [−3, 1]] = [[−2, 1], [1.5, −0.5]].
Conditions
Determinant and trace need a square matrix. A matrix with determinant 0 is singular and has no inverse. Check by multiplying the matrix by its inverse to get the identity.
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" shows "Every row needs the same number of numeric entries." beside that field and no result.
Choosing between the Matrix 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 Determinant Calculator](/math/matrix-determinant-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
- Determinant and trace need a square matrix
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 trace?
The sum of the diagonal entries.
What does a zero determinant mean?
The matrix is not invertible.
Can it multiply matrices?
Use the Matrix Multiplication calculator.
What does the Matrix Calculator calculate?
Determinant, trace, transpose and 2×2 inverse of a matrix. It returns Determinant, Trace, Transpose and Inverse.
What formula does the Matrix Calculator use?
det, trace, transpose; inverse (2×2) = (1/det)[[d, −b], [−c, a]]. Inverse is provided for 2×2 matrices with non-zero determinant.
What do I need to enter in the matrix 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. Determinant and trace need a square matrix.
What is an example of a matrix 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 Determinant Calculator — Determinant of a square matrix (up to 6×6).
- 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.