-3/4. Empty cells count as 0. Press Enter in any cell to calculate.To solve a system Ax = b, the solver writes the augmented matrix [A | b] and applies three kinds of row operations: swapping two rows, multiplying a row by a nonzero number, and adding a multiple of one row to another. These operations never change the solution set. The goal is the reduced row echelon form (RREF), where every pivot is 1 and is the only nonzero entry in its column.
If a row reduces to 0 = c with c ≠ 0, the system is inconsistent and has no solution. Otherwise, if every unknown has a pivot, the solution is unique. If some columns have no pivot, those unknowns are free variables and the system has infinitely many solutions, written in parametric vector form.
The determinant is computed by cofactor expansion (for 2×2 and 3×3) and by row reduction to upper-triangular form, where it equals the product of the diagonal times (−1) for each row swap. A square matrix is invertible exactly when its determinant is nonzero; its inverse is found by row-reducing [A | I] to [I | A⁻¹]. The rank is the number of pivots, and the nullity is the number of free columns.
Eigenvalues λ are the roots of the characteristic polynomial det(λI − A) = 0. For 2×2 matrices the graph shows how A transforms the unit square and draws the real eigenvector directions, which A only stretches and never rotates.