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. After reducing to triangular form, the determinant equals the product of the diagonal (each row swap flips the sign).

The inverse matrix is found by Gauss–Jordan: write [A | E] side by side and transform rows until the left part becomes the identity — the right part is then A⁻¹. 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.