Elementary matrix

Elementary row operations

Row-switching transformations

Properties

• The inverse of this matrix is itself: T_ij⁻¹ = T_ij.
• Since the determinant of the identity matrix is unity, det = −1. It follows that for any square matrix A, we have det = −det.

  Row-multiplying transformations

Properties

• The inverse of this matrix is given by D_i⁻¹ = D_i.
• The matrix and its inverse are diagonal matrices.
• det = m. Therefore for a square matrix A, we have det = m det.

  Row-addition transformations

Properties

• These transformations are a kind of shear mapping, also known as a transvections.
• The inverse of this matrix is given by L_ij⁻¹ = L_ij.
• The matrix and its inverse are triangular matrices.
• det = 1. Therefore, for a square matrix A we have det = det.
• Row-addition transforms satisfy the Steinberg relations.