Matrix Calculator
Enter a 2×2 or 3×3 matrix and get its determinant, inverse, transpose and trace — with a note on whether it can be inverted at all.
Type the matrix with a semicolon or a new line between rows and spaces or commas between entries, like "1 2; 3 4".
How the matrix calculator works
Type the matrix with a semicolon or a new line between rows and spaces or commas between entries, like "1 2; 3 4". The determinant of a 2×2 is ad − bc; for a 3×3 it expands along the first row. A matrix has an inverse only when the determinant is not zero — such a matrix is called singular, and it squashes space flat so the mapping cannot be undone. The inverse is the adjugate divided by the determinant.
Formula: det [a b; c d] = ad − bc; A⁻¹ = adj(A) / det(A)
Worked examples
| Inputs | Determinant | Note |
|---|---|---|
| [1 2; 3 4] | -2 | determinant −2 |
| [1 2 3; 4 5 6; 7 8 10] | -3 | determinant −3 |
| a singular matrix [1 2; 2 4] | 0 | determinant 0 — no inverse |
FAQFrequently asked questions
How do I type a matrix?
Rows separated by a semicolon or a new line, entries by spaces or commas: "1 2; 3 4" is a 2×2.
What does the determinant mean?
It is the factor by which the matrix scales area (2×2) or volume (3×3). A determinant of zero means everything is flattened, so the mapping cannot be reversed.
Why does my matrix have no inverse?
Because its determinant is zero — its rows are multiples of each other. Such a matrix is called singular.
What is the trace?
The sum of the diagonal entries. It equals the sum of the eigenvalues.
Can it do bigger matrices?
Not yet — 2×2 and 3×3 cover school and first-year university work; larger matrices are normally solved by elimination.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.