This matrix calculator works with 2 × 2 and 3 × 3 matrices. Find the determinant, inverse or transpose of a matrix, or add, subtract and multiply two matrices.
| Matrix A, column 1 | A, column 2 | A, column 3 (3 × 3 only) |
|---|---|---|
| Matrix B, column 1 | B, column 2 | B, column 3 (3 × 3 only) |
|---|---|---|
Key formulas
Determinant of a 2 × 2 matrix [[a, b], [c, d]]: ad − bc.
Inverse of a 2 × 2 matrix: (1 ÷ (ad − bc)) × [[d, −b], [−c, a]].
3 × 3 determinant: expand along the first row: a(ei − fh) − b(di − fg) + c(dh − eg). The inverse is the adjugate (transposed cofactor matrix) divided by the determinant.
Multiplication: each entry of A × B is the sum of a row of A multiplied term by term with a column of B.
Worked example
For A = [[2, 1], [1, 3]], the determinant is 2 × 3 − 1 × 1 = 5, and the inverse is (1/5) × [[3, −1], [−1, 2]] = [[0.6, −0.2], [−0.2, 0.4]].
When does a matrix have no inverse?
A matrix with a determinant of zero is singular: its rows (or columns) depend on each other, so it cannot be inverted. In a system of equations, this means there is no single unique solution.
Frequently asked questions
Is A × B the same as B × A?
Usually not. Matrix multiplication is not commutative, so the order matters.
What is the transpose of a matrix?
The matrix flipped over its diagonal, so rows become columns. Entry (i, j) moves to (j, i).
What is the identity matrix?
A square matrix with 1s on the diagonal and 0s elsewhere. Multiplying any matrix by it leaves the matrix unchanged, and a matrix times its inverse equals the identity.
Where are matrices used?
In solving systems of equations, computer graphics, physics, statistics, economics and machine learning.
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Last reviewed: October 2026.