Math 3321
If a homogeneous system of n linear equations in n unknowns has infinitely many solutions, then the matrix of coefficients is singular.
Always true
If a system of n linear equations in n unknowns is dependent, then the matrix of coefficients has rank less than or equal to n − 1.
Always true
If a system of n linear equations in n unknowns is inconsistent, then the reduced row echelon form of the matrix of coefficients is not In. (Letter I with subscript ''n'')
Always true
If the matrix of coefficients of a homogeneous system of n linear equations in n unknowns has an inverse, then the system does not have infinitely many solutions.
Always true
If the matrix of coefficients of a homogeneous system of n linear equations in n unknowns is nonsingular, then the trivial solution is the only solution of the system.
Always true
The rank of the matrix of coefficients of a homogeneous system of m linear equations in n unknowns is never less than the rank of the augmented matrix.
Always true
If 0 is an eigenvalue of the matrix of coefficients of a homogeneous system of n linear equations in n unknowns, then the system has infinitely many solutions.
Always true.
If 0 is not an eigenvalue of the matrix of coefficients of a homogeneous system of n linear equations in n unknowns, then the system does not have infinitely many solutions.
Always true.
If a system of n linear equations in n unknowns is dependent, then 0 is an eigenvalue of the matrix of coefficients.
Always true.
If the matrix of coefficients of a system of n linear equations in n unknowns is nonsingular, then the system has infinitely many solutions.
Never true,
If the reduced row echelon form of the matrix of coefficients of a system of n linear equations in n unknowns is not the identity matrix, then the determinant of the matrix of coefficients is non-zero
Never true,
If the rank of the augmented matrix of a system of n linear equations in n unknowns is greater than the rank of the matrix of coefficients, then the matrix of coefficients is nonsingular.
Never true, i.e., false
If a system of n linear equations in n unknowns is consistent, then the rank of the matrix of coefficients is n.
Sometimes true
If the matrix of coefficients of a system of n linear equations in n unknowns does not have an inverse, then the system has no solutions.
Sometimes true
If the reduced row echelon form of the matrix of coefficients of a system of n linear equations in n unknowns is not the identity, then the system is inconsistent.
Sometimes true
If 0 is an eigenvalue of the matrix of coefficients of a system of n linear equations in n unknowns, then the system has infinitely many solutions.
Sometimes true.
If a system of n linear equations in n unknowns has infinitely many solutions, then the rank of the matrix of coefficients is n − 1.
Sometimes true.
