LINEAR ALGEBRA DEFINITIONS AND
THEOREMS EXAM QUESTIONS WITH
CORRECT ANSWERS
Linear Equation - Answer-A linear equation in the variables x 1, ••. , x 11 is an equation
that can be written in the form
a1x1 +a2x2.... anxn
System of Linear Equations - Answer-A system of linear equations (or a linear system)
is a collection of one or more linear equations involving the same variables-say, x1, .. • ,
x,,.
Consistent - Answer-A system of linear equations is said to be consistent if it has either
one solution or infinitely many solutions;
Inconsistent - Answer-a system is inconsistent if it has no solution.
Elementary Row Operations - Answer-1. (Replacement) Replace one row by the sum of
itself and a multiple of another row. 1
2. (Interchange) Interchange two rows.
3. (Scaling) Multiply all entries in a row by a nonzero constant.
TWO FUNDAMENTAL QUESTIONS ABOUT A LINEAR SYSTEM - Answer-1. Is the
system consistent; that is, does at least one solution exist?
2. If a solution exists, is it the only one; that is, is the solution unique?
Properties of a Matrix in Echelon Form/Row Echelon Form - Answer-A rectangular
matrix is in echelon form (or row echelon form) if it has the following three properties:
1. All nonzero rows are above any rows of all zeros.
2. Each leading 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 zeros.
Properties of a Matrix in Reduced Echelon Form/ Reduced Row Echelon Form -
Answer-If a matrix in echelon form satisfies the following additional conditions, then it is
in reduced echelon form (or reduced row echelon form):
1. All nonzero rows are above any rows of all zeros.
2. Each leading 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 zeros.
4. The leading entry in each nonzero row is 1.
S. Each leading 1 is the only nonzero entry in its column.
THEOREMS EXAM QUESTIONS WITH
CORRECT ANSWERS
Linear Equation - Answer-A linear equation in the variables x 1, ••. , x 11 is an equation
that can be written in the form
a1x1 +a2x2.... anxn
System of Linear Equations - Answer-A system of linear equations (or a linear system)
is a collection of one or more linear equations involving the same variables-say, x1, .. • ,
x,,.
Consistent - Answer-A system of linear equations is said to be consistent if it has either
one solution or infinitely many solutions;
Inconsistent - Answer-a system is inconsistent if it has no solution.
Elementary Row Operations - Answer-1. (Replacement) Replace one row by the sum of
itself and a multiple of another row. 1
2. (Interchange) Interchange two rows.
3. (Scaling) Multiply all entries in a row by a nonzero constant.
TWO FUNDAMENTAL QUESTIONS ABOUT A LINEAR SYSTEM - Answer-1. Is the
system consistent; that is, does at least one solution exist?
2. If a solution exists, is it the only one; that is, is the solution unique?
Properties of a Matrix in Echelon Form/Row Echelon Form - Answer-A rectangular
matrix is in echelon form (or row echelon form) if it has the following three properties:
1. All nonzero rows are above any rows of all zeros.
2. Each leading 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 zeros.
Properties of a Matrix in Reduced Echelon Form/ Reduced Row Echelon Form -
Answer-If a matrix in echelon form satisfies the following additional conditions, then it is
in reduced echelon form (or reduced row echelon form):
1. All nonzero rows are above any rows of all zeros.
2. Each leading 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 zeros.
4. The leading entry in each nonzero row is 1.
S. Each leading 1 is the only nonzero entry in its column.
