Matrix calculator
Explore the core properties of a 2×2 or 3×3 real matrix, including determinant, trace, rank, transpose, inverse, norm, and characteristic polynomial.
Matrix input
Choose one of the two square sizes supported by this matrix overview.
Enter ordinary real numbers such as 4, – 2.5, or 0.125. Comma decimals and scientific notation are intentionally rejected to avoid ambiguous input.
Live results
Matrix detail
| Row | Input A | Transpose Aᵀ | Inverse A⁻¹ |
|---|---|---|---|
| 1 | [1, 2] | [1, 3] | [ – 2, 1] |
| 2 | [3, 4] | [2, 4] | [1.5, – 0.5] |
How to use the matrix calculator
What this calculator does
This calculator summarizes the most useful basic properties of a real square matrix. It accepts either a 2 × 2 or 3 × 3 matrix and calculates its determinant, trace, rank, Frobenius norm, transpose, inverse when one exists, and characteristic polynomial. These quantities answer different questions: the determinant indicates singularity and scaling, the trace summarizes the main diagonal, the rank measures the number of independent rows or columns, and the inverse is the matrix that reverses multiplication by the original matrix. The tool is an arithmetic and learning aid; it does not decide whether a matrix model is appropriate for a real-world problem.
Use it when checking linear-algebra homework, verifying an intermediate matrix in a larger calculation, testing whether a small coefficient matrix is invertible before solving a system, or comparing hand-computed determinant and inverse work. For a broader introduction to rows, columns, entries, and matrix operations, see OpenStax on matrices and matrix operations.
How to calculate
- The calculator opens with the demonstration matrix A = [1, 2], [3, 4][1, 2], [3, 4][a, b], [c, d][d, – b], [ – c, a][ – 2, 1], [1.5, – 0.5][A | I], again with pivoting. These algorithms are convenient for small real matrices but, like all floating-point computations, can be sensitive when a matrix is extremely close to singular.
2 × 2: det(A) = ad – bc • tr(A) = a + d • ||A||F = √Σ aᵢⱼ²
Interpretation and common mistakes
- A small nonzero determinant does not by itself prove that a numerical problem is well-conditioned; it only establishes nonsingularity at the chosen scale and precision.
- Rank is not the number of nonzero entries. A matrix can contain many nonzero values while its rows are still linearly dependent.
- The transpose is not the inverse. They coincide only for special matrices such as orthogonal matrices.
- Do not round coefficients before entering them unless the original problem also uses those rounded values. Small entry changes can produce much larger changes in an inverse near singularity.