Matrix calculator

How it works

A matrix is a table of numbers, and most questions about it are answered by row operations: swap two rows, multiply a row by a number that can be undone, or subtract a multiple of one row from another. Gaussian elimination uses them to clear the entries below each pivot, which gives a row echelon form; continuing upwards and scaling every pivot to 1 gives the reduced row echelon form, which is unique. The rank is the number of pivots.

For the determinant, subtracting a multiple of a row changes nothing and a swap changes the sign, so after elimination $\det A$ is the product of the diagonal, with one minus sign per swap. For matrices up to 3 by 3 the cofactor expansion is shown as well, for example $\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$. For the inverse, Gauss-Jordan elimination is run on $[A \mid I]$: when the left half has become $I$, the right half is $A^{-1}$.

Modulo $n$, dividing by a pivot $a$ means multiplying by $a^{-1} \bmod n$, which exists only when $\gcd(a, n) = 1$. When $n$ is composite, a column can have non-zero entries none of which is invertible; the calculator then runs Euclid’s algorithm on that column with whole multiples of rows, which needs no division. A matrix is invertible modulo $n$ exactly when $\gcd(\det A, n) = 1$. This is the arithmetic of the Hill cipher, whose key is a matrix modulo 26.