Every invertible n×n matrix A can be inverted mechanically by row reduction: form the n×2n augmented matrix [A | I], apply elementary row operations until the left half becomes the identity, and the right half is then A⁻¹; if a row of zeros appears on the left, no inverse exists. This algorithm is the constructive counterpart to the determinant test for invertibility, and it rests on three equivalent interpretations of a linear system y = Ax: as a coefficient matrix, as a single matrix equation solved via A⁻¹y = x, and as a mapping f between n-dimensional spaces whose invertibility corresponds exactly to f being one-to-one and onto. This belongs to linear algebra, specifically the theory of matrix inversion, systems of linear equations, and linear mappings, forming a foundation for later work on functions of several variables.
Every invertible n×n matrix A can be inverted mechanically by row reduction: form the n×2n augmented matrix [A | I], apply elementary row operations until the left half becomes the identity, and the …