CH. 1.2- ROW REDUCTION AND
ECHELONS FORMS QUESTIONS &
VERIFIED CORRECT ANSWERS
Echelon form - Correct Answer ✔✔ 1) All nonzero rows are above any rows of all zeros.
2) Each leading entry (left most nonzero entry) of a row is in a column to the right of the
leading entry of the row above it.
3) All entries in a column below a leading entry are zero.
Reduced echelon form - Correct Answer ✔✔ 1) The leading entry in each nonzero row
is 1.
2) Each leading 1 is the only nonzero entry in its column.
Theorem 1: Uniqueness of the reduced echelon form - Correct Answer ✔✔ Each matrix
is row-equivalent to one and only one reduced echelon matrix.
Pivot position - Correct Answer ✔✔ A position of a leading entry in an echelon form of
the matrix.
Pivot - Correct Answer ✔✔ A nonzero number that either is used in a pivot position to
create 0'a or is changed into a leading 1, which in turn is used to create 0's.
Pivot column - Correct Answer ✔✔ A column that contains a pivot position.
Basic variable - Correct Answer ✔✔ Any variable that corresponds to a pivot column in
the augmented matrix of a system.
Free variable - Correct Answer ✔✔ All nonbasic variables.
General solution of the system - Correct Answer ✔✔ It provides a parametric
description of the solution set (free variables act as parameters).
Consistent system with free variables - Correct Answer ✔✔ Infinitely many solutions
Consistent system with no free variables - Correct Answer ✔✔ Unique solution: one &
only one solution
Theorem 2: Existence and Uniqueness Theorem - Correct Answer ✔✔ 1) A linear
system is consistent if only and only if the rightmost column of the augmented matrix is
not a pivot column, i.e. if only and only if an echelon form of the augmented matrix has
no row of the form [ 0...0 b] where b is nonzero.
2) If a linear system is consistent, then the solutions contains either
ECHELONS FORMS QUESTIONS &
VERIFIED CORRECT ANSWERS
Echelon form - Correct Answer ✔✔ 1) All nonzero rows are above any rows of all zeros.
2) Each leading entry (left most nonzero entry) of a row is in a column to the right of the
leading entry of the row above it.
3) All entries in a column below a leading entry are zero.
Reduced echelon form - Correct Answer ✔✔ 1) The leading entry in each nonzero row
is 1.
2) Each leading 1 is the only nonzero entry in its column.
Theorem 1: Uniqueness of the reduced echelon form - Correct Answer ✔✔ Each matrix
is row-equivalent to one and only one reduced echelon matrix.
Pivot position - Correct Answer ✔✔ A position of a leading entry in an echelon form of
the matrix.
Pivot - Correct Answer ✔✔ A nonzero number that either is used in a pivot position to
create 0'a or is changed into a leading 1, which in turn is used to create 0's.
Pivot column - Correct Answer ✔✔ A column that contains a pivot position.
Basic variable - Correct Answer ✔✔ Any variable that corresponds to a pivot column in
the augmented matrix of a system.
Free variable - Correct Answer ✔✔ All nonbasic variables.
General solution of the system - Correct Answer ✔✔ It provides a parametric
description of the solution set (free variables act as parameters).
Consistent system with free variables - Correct Answer ✔✔ Infinitely many solutions
Consistent system with no free variables - Correct Answer ✔✔ Unique solution: one &
only one solution
Theorem 2: Existence and Uniqueness Theorem - Correct Answer ✔✔ 1) A linear
system is consistent if only and only if the rightmost column of the augmented matrix is
not a pivot column, i.e. if only and only if an echelon form of the augmented matrix has
no row of the form [ 0...0 b] where b is nonzero.
2) If a linear system is consistent, then the solutions contains either
