Matrix Determinant & Inverse
Compute the determinant and inverse of 2×2 and 3×3 matrices with cofactor-expansion working. Detects singular matrices (det = 0) and verifies the inverse — useful for math B and linear algebra students.
Try an example
Enter the matrix entries to see its determinant and inverse.
Done
Matrix Determinant & Inverse
How to use
- 1
Pick the size
Choose a 2×2 or 3×3 matrix.
- 2
Enter the entries
Fill in each entry, or use one of the example buttons.
- 3
Read determinant and inverse
The determinant shows cofactor-expansion working, and the inverse (when it exists) comes with a verification.
Features
- 2×2 and 3×3 determinants with cofactor-expansion working
- Inverse shown as a grid, with the (1/det) formula steps for 2×2
- Automatic detection and notice of singular matrices (det = 0)
- A × A⁻¹ verification included; inputs are never transmitted
Use cases
Check matrix homework
Compare hand-worked determinants and inverses with the shown working.
Prep for linear systems
Confirm a matrix is invertible (det ≠ 0) before matrix-method solutions.
Inspect transformations
Check the determinant of rotation or scaling matrices — the area factor.
Details
A 2×2 determinant is det = a·d − b·c; a 3×3 determinant expands along the first row as det = a(ei − fh) − b(di − fg) + c(dh − eg). The tool substitutes your entries into these small-matrix formulas step by step, so the working doubles as an answer key.
An inverse exists only when det ≠ 0. For 2×2 it is (1/det) × [d, −b; −c, a]; for 3×3 each entry is a cofactor divided by the determinant, displayed as a grid. A matrix with det = 0 is singular, and the tool states that clearly instead of producing numbers.
The inverse is verified by multiplying it with the original matrix — the product should be the identity — and the tool shows the first diagonal entry of that product for 3×3. Entries accept decimals (results rounded to six places) with absolute values up to one million. Inputs are never transmitted; everything is computed locally in your browser.
FAQ
What is a determinant?
It is a single number computed from a square matrix — for 2×2, a·d − b·c. Geometrically it is the factor by which the matrix stretches space; when it is 0, space collapses and no inverse exists.
When does an inverse not exist?
When the determinant is 0. This corresponds to the associated linear system having no unique solution (like two parallel lines), and the tool reports "no inverse" in that case.
Does the 3×3 expansion work along any row or column?
Yes — expanding along any row or column yields the same value. The tool expands along the first row; picking a row or column with many zeros makes hand calculation easier.
Can I use decimal entries?
Yes. Results are rounded to six decimal places. If you need exact fraction forms, compute with integer entries first.
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Verified: Known values (det −2 with inverse, 3×3 det −1, singular detection), decimal and out-of-range inputs and the English page are covered by browser tests
Did you know?
The determinant of any rotation matrix (say, 30°) is always exactly 1 — rotations preserve shape and area, a clean illustration that a determinant measures area scaling.