Matrix Calculator

How to use the matrix calculator

Pick the operation, set the number of rows and columns, and type the entries. Empty cells count as 0, and you can enter decimals or fractions such as 3/4. Every result is calculated in exact fractions, so 1/3 stays 1/3 instead of 0.3333 – switch to Decimals if you prefer.

Adding and subtracting matrices

Two matrices can be added only if they have the same size. You add the entries in the same position: the entry in row 1, column 2 of A + B is a₁₂ + b₁₂. Subtraction works the same way.

Multiplying matrices

A × B exists when A has as many columns as B has rows. A 2×3 matrix times a 3×4 matrix gives a 2×4 matrix. Each entry is a row of A multiplied by a column of B, entry by entry, then added up:

c_ij = a_i1·b_1j + a_i2·b_2j + … + a_in·b_nj

Example: A = [2 1; 1 3] and B = [1 2; 0 1]. The top-left entry is 2·1 + 1·0 = 2, the top-right 2·2 + 1·1 = 5, so A × B = [2 5; 1 5]. Note that matrix multiplication is not commutative: here B × A = [4 7; 1 3], a different matrix.

Other operations

• k · A – every entry is multiplied by the number k.
• Transpose Aᵀ – rows become columns; a 2×3 matrix becomes 3×2.
• Power Aⁿ – A multiplied by itself n times (square matrices only); A⁰ is the identity matrix I.
• Determinant – a single number; the matrix has an inverse exactly when it is not 0.
• Inverse A⁻¹ – the matrix with A × A⁻¹ = I.
• Rank – the number of linearly independent rows (or columns).
• Trace – the sum of the entries on the main diagonal.

Which operations need a square matrix?

The determinant, the inverse, powers and the trace are defined only for square matrices (2×2, 3×3 …). Addition, multiplication, the transpose and the rank work for any size, as long as the dimensions match.

Frequently asked questions