Matrix Calculator
Determinant, trace, transpose and inverse of a 2×2 or 3×3 matrix, with the cofactor expansion written out and singular matrices called out explicitly.
Matrix Calculator: with the default inputs, determinant is 10.
A 2×2 uses only the top-left four entries.
- Trace (sum of the diagonal)
- 10
- Singular?
- No — the determinant is not 0, so the matrix is invertible
- Inverse
- [0.6 -0.7; -0.2 0.4]
- Transpose
- [4 2; 7 6]
Assumptions
- Only real-valued 2×2 and 3×3 matrices are supported.
- The inverse is computed from the adjugate divided by the determinant — exact for exact inputs, but ill-conditioned matrices amplify rounding error.
- A determinant of exactly 0 is reported as singular; a very small non-zero determinant is still inverted, with the usual numerical caveats.
| Column 1 | Column 2 |
|---|---|
| 4 | 7 |
| 2 | 6 |
| Column 1 | Column 2 |
|---|---|
| 0.6 | -0.7 |
| -0.2 | 0.4 |
Multiply the matrix by this and you get the identity, up to rounding.
How this is worked out
The formula
2×2: det = a₁₁a₂₂ − a₁₂a₂₁
inverse = (1 ÷ det) × [ a₂₂ −a₁₂ ; −a₂₁ a₁₁ ]
3×3: det = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁)
inverse = adjugate ÷ det, where the adjugate is the transpose of the cofactor matrix
Trace = sum of the diagonal entries. A matrix is singular exactly when det = 0.Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Matrix size
- A 2×2 uses only the top-left four entries.2 × 2 · 3 × 3
- a11
- A number.defaults to 4
- a12
- A number.defaults to 7
- a13(under More options)
- A number.defaults to 2
- a21
- A number.defaults to 2
- a22
- A number.defaults to 6
- a23(under More options)
- A number.defaults to 3
- a31(under More options)
- A number.defaults to 1
- a32(under More options)
- A number.defaults to 2
- a33(under More options)
- A number.defaults to 9
What you get back
- Determinantmain answer
- Trace (sum of the diagonal)
- Singular?
- Inverse
- Transpose
What this assumes
- Only real-valued 2×2 and 3×3 matrices are supported.
- The inverse is computed from the adjugate divided by the determinant — exact for exact inputs, but ill-conditioned matrices amplify rounding error.
- A determinant of exactly 0 is reported as singular; a very small non-zero determinant is still inverted, with the usual numerical caveats.
About this calculator
The determinant is the single number that tells you the most about a square matrix. It is the factor by which the matrix scales area (2×2) or volume (3×3), and its sign says whether the transformation flips orientation. A determinant of 0 means the matrix squashes space flat — and that is precisely when no inverse exists.
How to use it
Choose 2×2 or 3×3 and fill in the entries; a 2×2 uses only the top-left four boxes, and the third row and column are tucked under More options. The steps show the cofactor expansion in full, which is the working a linear algebra course expects to see.
How to read the results
- Determinant. For a 2×2 matrix built from the vectors (a₁₁, a₂₁) and (a₁₂, a₂₂), |det| is the area of the parallelogram they span. For a 3×3 it is the volume of the parallelepiped. A negative determinant means the transformation reverses handedness (a reflection is mixed in).
- Trace is the sum of the diagonal. It equals the sum of the eigenvalues, just as the determinant equals their product — the two numbers together pin down a 2×2 matrix's characteristic polynomial completely.
- Inverse. Computed from the adjugate divided by the determinant, which is the exact closed form rather than a numerical elimination. Multiply the original by the inverse and you get the identity matrix.
- Transpose flips rows and columns. Note that det(Aᵀ) = det(A) always.
Singular matrices
When the determinant is exactly 0 the matrix is singular and there is no inverse. The calculator says so explicitly rather than printing infinities. In practice this means the rows (equivalently the columns) are linearly dependent — one is a combination of the others — and the system Ax = b either has no solution or infinitely many, never exactly one.
Where it misleads
A determinant that is merely small is a warning sign of its own. The matrix [[1, 1], [1, 1.0001]] has a determinant of 0.0001 and a perfectly well-defined inverse, but that inverse has entries in the tens of thousands, so tiny errors in the input become enormous errors in the output. Numerical analysts call this ill-conditioning, and it is why serious software solves linear systems by LU decomposition with pivoting rather than by forming the inverse at all. The closed-form inverse here is exact for exact inputs and ideal for teaching; for solving a large or delicate system, use a proper solver.
Frequently asked questions
▸How do I find the determinant of a 2×2 matrix?
Multiply the main diagonal and subtract the product of the other diagonal: ad − bc. For [[1, 2], [3, 4]] that is 1×4 − 2×3 = −2.
▸What does it mean when the determinant is zero?
The matrix is singular. It maps space onto a lower-dimensional subspace, has no inverse, and the associated system of equations has either no solution or infinitely many.
▸How do you invert a 3×3 matrix by hand?
Build the 3×3 matrix of cofactors, transpose it to get the adjugate, then divide every entry by the determinant. The calculator does exactly that, so the steps match a textbook worked example.
▸Is the determinant of a transpose the same?
Yes — det(Aᵀ) = det(A) for every square matrix. Swapping rows and columns never changes the determinant.
▸Why can't I do 4×4 or matrix multiplication here?
This calculator is built around the closed-form 2×2 and 3×3 results that are worth showing the working for. Larger matrices need elimination-based methods where the useful output is the algorithm's trace, not a formula.
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