Matrix Calculator

Add, subtract, multiply, transpose, or find the determinant of matrices up to 4×4

Matrix operations — addition, subtraction, multiplication, transpose, determinant — each follow distinct rules that don't generalize from ordinary number arithmetic, and matrix multiplication in particular behaves in ways that surprise people expecting it to work like multiplying regular numbers.

Why matrix multiplication isn't just "multiply corresponding entries"

Matrix addition and subtraction work element-by-element, similar to ordinary arithmetic intuition — but matrix multiplication follows an entirely different rule, computing each entry of the result as a sum of products across a row of one matrix and a column of the other, not simply multiplying corresponding positions together. This has real consequences: matrix multiplication is generally not commutative (A×B usually doesn't equal B×A, unlike ordinary number multiplication where order never matters), and matrix dimensions have to be compatible in a specific way (the number of columns in the first matrix must match the number of rows in the second) for multiplication to even be defined at all. The determinant, meanwhile, is a single number calculated from a square matrix that captures important properties about the matrix — including whether the matrix is invertible (a zero determinant means it isn't).

A worked example

Multiplying two matrices in one order versus the reverse order can produce two completely different results (or one order might not even be mathematically valid if the dimensions don't align that way) — this non-commutative behavior is exactly why matrix multiplication order matters in a way that ordinary number multiplication order never does, and it's one of the first things that trips people up moving from arithmetic to matrix algebra.

How this connects to your other math calculations

Matrix operations underpin a wide range of applications from computer graphics transformations to systems of linear equations — for the more basic algebraic operations that matrices build on conceptually, the Exponent Calculator and other algebra tools on this site cover the foundational operations, though matrix algebra itself follows its own distinct rule set beyond simple arithmetic extension.

Common mistakes

Assuming matrix multiplication works like element-by-element multiplication (multiplying each position in one matrix by the corresponding position in the other) rather than the actual row-by-column sum-of-products rule is the most common conceptual error — the two operations produce genuinely different results, and only the row-by-column method is mathematically correct matrix multiplication.