Matrix calculator — every step written the way you would write it on paper

Gaussian elimination is the main technique for matrices. Three elementary row operations are allowed: swap two rows, multiply a row by a non-zero number, and add a multiple of one row to another. They do not change the solutions of the system.

The determinant is a number that shows whether the matrix is degenerate. If det A = 0, the matrix has no inverse and the system A·x = b has no unique solution. For 2×2, det = a₁₁·a₂₂ − a₁₂·a₂₁; larger matrices are expanded along the first row: each of its elements is multiplied by its cofactor Aᵢⱼ = (−1)ⁱ⁺ʲ·Mᵢⱼ, where the minor Mᵢⱼ is the determinant of the matrix with row i and column j deleted.

The inverse matrix is computed by the formula A⁻¹ = (1/det A)·adj A: find the cofactors of all elements, write them into a matrix, transpose it (this gives the adjugate matrix adj A) and divide every element by the determinant. Check: A·A⁻¹ = E.

The rank is the number of non-zero rows in row echelon form, i.e. how many equations are truly independent.

Matrix multiplication: the element cᵢⱼ is the dot product of row i of the first matrix and column j of the second. That is why the number of columns of A must equal the number of rows of B, and in general A×B ≠ B×A.

Cells accept integers, decimals (1.5) and fractions (2/3) — all computations use exact fractions, no rounding errors.