Matrix Calculator
Perform matrix operations on 2×2, 2×3, or 3×3 matrices. Add, subtract, multiply, transpose and compute determinants with the full working shown so you can trace each step. Ideal for linear algebra homework and quick checks.
Configure matrices and operation
Enter matrices and select an operation.
Matrix operations
Addition and subtraction
Add or subtract matrices element-by-element. Both matrices must have the same dimensions.
(A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ
Multiplication
For 2×2 matrices, each entry of the product is the dot product of a row of A and a column of B.
(A × B)ᵢⱼ = Σₖ Aᵢₖ · Bₖⱼ
Transpose
The transpose flips the matrix across its main diagonal — rows become columns and columns become rows.
(Aᵀ)ᵢⱼ = Aⱼᵢ
Determinant
det [[a, b], [c, d]] = ad − bc
For 3×3 matrices the determinant is computed using cofactor expansion along the first row:
det A = a(ei − fh) − b(di − fg) + c(dh − eg)
Worked examples
Example 1 — Matrix addition
[[1,2],[3,4]] + [[5,6],[7,8]] = [[6,8],[10,12]]
Result: [[6,8],[10,12]]
Example 2 — Determinant
det [[2,3],[1,4]] = 2×4 − 3×1 = 8 − 3 = 5
det = 5
Example 3 — Multiplication
[[1,2],[3,4]] × [[5,6],[7,8]] = [[19,22],[43,50]]
Result: [[19,22],[43,50]]
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Frequently asked questions
Multiplication is defined when the number of columns in A equals the number of rows in B. On this page, both matrices are square (2×2 or 3×3) so multiplication always works.
A zero determinant means the matrix is singular — it has no inverse, and its rows (or columns) are linearly dependent. Geometrically, it represents a transformation that collapses the plane onto a line or point.
No. In general A × B ≠ B × A. The order of multiplication matters, unlike scalar multiplication.