Thus the categorization is somewhat arbitrary. The facts within each category are also numbered arbitrarily. A homogeneous system of linear equations is consistent . An underdetermined homogeneous system has infinitely many solutions. A consistent linear system has exactly one solution if and only if it has no free variables .
Every matrix is row equivalent to one and only one matrix in reduced row echelon form .
A the following are equivalent. A has a leading 1 in every row. If the coefficient matrix for a system of equations is diagonally dominant then the Jacobi and Gauss-Seidel iterations will converge. If A and B are similar matrices then they have the same eigenvalues . A square matrix is invertible iff it is a product of elementary matrices .