A matrix is an ordered rectangle of values. Its rows and columns are part of its meaning, so operations must respect position and dimensions.
Use it to compare the distinct rules for adding, subtracting, multiplying, or multiplying entry by entry.
The relationship
The defining relationship
A+B, A−B, AB, or A⊙B
See the structure
What the calculation is doing
1234
same positions
5678
A+B, A−B, AB, or A⊙B
Worked interpretation
Read the result in context
A+B, A−B, AB, or A⊙B
Interpret with care
Important boundary
Matrix multiplication is not entrywise and generally AB ≠ BA.
Use the Matrix Calculator to experiment with small exact matrix operations before applying a more specialized route.
Quick guide
How to use this calculator
Choose an operation.
Set the dimensions of matrices A and B.
Fill every entry and read the exact result matrix.
Calculation method
Use the rule selected for A and B
Addition, subtraction, and the Hadamard product combine matching entries. Matrix multiplication instead takes dot products of rows of A with columns of B, so order and dimensions matter.
Worked example
Add two 2 × 2 matrices
For A = [[1,2],[3,4]] and B = [[5,6],[7,8]], add corresponding entries.
A + B = [[6,8],[10,12]]
Supported inputs
Precision and limits
Matrix size
Each input matrix may contain 1–6 rows and 1–6 columns.
Entry precision
Each signed decimal may contain up to 30 digits and 15 decimal places.
Exactness
Finite decimals are converted to reduced fractions before calculation; matrix arithmetic is not rounded.
