Matrix Operations
Matrix calculator: addition and subtraction, scalar and matrix multiplication, transpose and matrix powers with a step-by-step solution in fractions.
What to find:
Entries are whole numbers, decimals or fractions: 3, −1.5, 2/3. Size up to 6×6.
The matrix operations calculator adds and subtracts matrices, multiplies a matrix by a number and by another matrix, transposes a matrix and raises a square matrix to a power. Entries can be fractions — the calculation is exact, and every entry of the result is written out.
Rules
| Operation | Rule | Condition |
|---|---|---|
| Sum, difference | (A ± B)ij = aij ± bij | same size |
| Scalar multiplication | (kA)ij = k · aij | any matrix |
| Product | (AB)ij = ai1b1j + ai2b2j + … + ainbnj | columns of A = rows of B |
| Transpose | (AT)ij = aji | any matrix |
| Power | An = A · A · … · A | square matrix |
Example. A 3×2 matrix times a 2×3 matrix is a 3×3 matrix:
c11 = 1 · 1 + 2 · (−1) = −1, c12 = 1 · 0 + 2 · 3 = 6, and so on.
Properties of matrix multiplication
- The product is generally not commutative: AB ≠ BA.
- Associative and distributive: (AB)C = A(BC), A(B + C) = AB + AC.
- The identity matrix E does not change a matrix: AE = EA = A.
- Transpose of a product: (AB)T = BTAT.
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