Matrix Calculator
How Does This Calculator Work?
This calculator performs basic matrix operations based on the rules of linear algebra:
- Addition & Subtraction: To add or subtract two matrices, they must have the exact same dimensions. The operation is performed element-wise, meaning the elements in the same row and column position are added or subtracted.
- Multiplication: To multiply matrix A by matrix B, the number of columns in A must equal the number of rows in B. The element in the i-th row and j-th column of the result is calculated by taking the dot product of the i-th row of A and the j-th column of B.
The Surprising History of Matrix Algebra
Unlike concepts like geometry or fractions, which are thousands of years old, matrix algebra is a relatively modern invention from the mid-19th century. The idea of arranging numbers in a grid to solve systems of linear equations existed earlier, but the concept of a "matrix" as a distinct mathematical object was not formalized.
The credit for this revolution goes to the English mathematician Arthur Cayley. In his 1858 "Memoir on the Theory of Matrices," he was the first to define matrices and their algebra, including addition, subtraction, multiplication, and inversion. What's surprising is that for a long time, matrix theory was considered a niche, abstract area of pure mathematics with few practical applications. It wasn't until the 20th century that its true power was unleashed, becoming an essential tool in quantum mechanics, computer graphics (for 3D transformations), economics, and modern data science.
Explore More Related Tools
While you're here, check out some of our other popular math and science calculators:
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Frequently Asked Questions (FAQ)
Why can't I multiply two matrices?
The most common reason for a multiplication error is that the dimensions are incompatible. For the multiplication A × B to be valid, the number of columns in Matrix A must be exactly equal to the number of rows in Matrix B. Our calculator automatically syncs these dimensions for you to prevent errors.
Is A × B the same as B × A?
No. Unlike regular number multiplication, matrix multiplication is generally not commutative. A × B will usually produce a different result from B × A. In fact, sometimes B × A may not even be a valid operation if the inner dimensions don't match.
What are matrices used for?
Matrices are incredibly powerful tools used in many fields. They are used to solve systems of linear equations, in computer graphics to rotate and scale 3D models, in physics to describe quantum states, and in data science and machine learning to represent and process large datasets.