WGU E082 – Linear Algebra for
Engineers: 100-Question Exam with
correct answers and rationale
updated 2026 graded A+
Section 1: Systems of Linear Equations (Questions 1–
15)
1. Which of the following is the augmented matrix for the system:
2x + y = 5
x - 3y = -1
A. [2 1 | 5; 1 -3 | -1]
B. [2 1; 1 -3]
C. [5 1 | 2; -1 -3 | 1]
D. [2 1 | -1; 1 -3 | 5]
Answer: A
Rationale: The augmented matrix places coefficients of variables in columns and the
constants in an additional column separated by a vertical bar. The first row
corresponds to 2x + y = 5, and the second to x - 3y = -1.
2. A system of linear equations is consistent if and only if:
A. The number of equations equals the number of variables.
B. The rightmost column of the augmented matrix is not a pivot column.
C. The coefficient matrix has full rank.
D. The determinant of the coefficient matrix is nonzero.
Answer: B
Rationale: By the Existence and Uniqueness Theorem, a linear system is consistent if
and only if the rightmost column of the augmented matrix is not a pivot column—
that is, there is no row of the form [0 0 ... 0 | b] with b ≠ 0.
3. How many solutions does the following system have?
x + 2y = 3
2x + 4y = 6
A. No solution
B. Exactly one solution
,C. Infinitely many solutions
D. Exactly two solutions
Answer: C
Rationale: The second equation is a multiple of the first (2×), so the two lines are
identical. Since there is one free variable, the system has infinitely many solutions.
4. Which operation is NOT an elementary row operation?
A. Interchanging two rows
B. Multiplying a row by a nonzero scalar
C. Replacing a row by the sum of that row and a multiple of another row
D. Multiplying two rows together
Answer: D
Rationale: The three elementary row operations are row replacement, row scaling
(by a nonzero constant), and row interchange. Multiplying entire rows together is not
a valid elementary operation.
5. The reduced row echelon form (RREF) of a matrix is unique because:
A. The algorithm always produces the same result regardless of row operation order.
B. All matrices can be reduced to the identity matrix.
C. Row operations are commutative.
D. The determinant determines the RREF.
Answer: A
Rationale: The RREF is unique—a theorem in linear algebra states that every matrix
is row equivalent to exactly one reduced row echelon form, regardless of the
sequence of row operations used.
6. For a system Ax = b, a particular solution x_p plus any vector from the null
space of A produces:
A. A particular solution
B. The general solution
C. The trivial solution
D. An inconsistent system
Answer: B
Rationale: If x_p is one solution to Ax = b and v is any solution to Ax = 0 (the
homogeneous system), then x_p + v is also a solution. The set of all solutions is x_p +
Null(A).
7. The homogeneous system Ax = 0 always has:
, A. No solution
B. The trivial solution x = 0
C. Infinitely many solutions
D. A unique nontrivial solution
Answer: B
Rationale: Setting all variables to zero always satisfies Ax = 0. A nontrivial solution
exists if and only if there is at least one free variable.
8. If a system has more variables than equations, then:
A. The system is always inconsistent.
B. The system has a unique solution.
C. The system has either no solution or infinitely many solutions.
D. The system is always consistent.
Answer: C
Rationale: With more variables than equations, there cannot be a unique solution.
The system is either inconsistent (no solution) or consistent with at least one free
variable (infinitely many solutions).
9. A pivot position in a matrix is:
A. Any nonzero entry
B. A location that corresponds to a leading 1 in RREF
C. Always in the last column
D. The largest entry in a row
Answer: B
Rationale: A pivot position is a location in the matrix that corresponds to a leading
entry (pivot) in the row echelon form or RREF. Pivot columns are columns containing
pivot positions.
10. The dimension of the solution set of a consistent system equals:
A. The number of pivot columns
B. The number of free variables
C. The number of equations
D. The rank of the coefficient matrix
Answer: B
Rationale: If a consistent system has n variables and p pivot columns, there are n − p
free variables. The solution set is a translated subspace whose dimension equals the
number of free variables.
11. In the equation Ax = b, the matrix A represents:
Engineers: 100-Question Exam with
correct answers and rationale
updated 2026 graded A+
Section 1: Systems of Linear Equations (Questions 1–
15)
1. Which of the following is the augmented matrix for the system:
2x + y = 5
x - 3y = -1
A. [2 1 | 5; 1 -3 | -1]
B. [2 1; 1 -3]
C. [5 1 | 2; -1 -3 | 1]
D. [2 1 | -1; 1 -3 | 5]
Answer: A
Rationale: The augmented matrix places coefficients of variables in columns and the
constants in an additional column separated by a vertical bar. The first row
corresponds to 2x + y = 5, and the second to x - 3y = -1.
2. A system of linear equations is consistent if and only if:
A. The number of equations equals the number of variables.
B. The rightmost column of the augmented matrix is not a pivot column.
C. The coefficient matrix has full rank.
D. The determinant of the coefficient matrix is nonzero.
Answer: B
Rationale: By the Existence and Uniqueness Theorem, a linear system is consistent if
and only if the rightmost column of the augmented matrix is not a pivot column—
that is, there is no row of the form [0 0 ... 0 | b] with b ≠ 0.
3. How many solutions does the following system have?
x + 2y = 3
2x + 4y = 6
A. No solution
B. Exactly one solution
,C. Infinitely many solutions
D. Exactly two solutions
Answer: C
Rationale: The second equation is a multiple of the first (2×), so the two lines are
identical. Since there is one free variable, the system has infinitely many solutions.
4. Which operation is NOT an elementary row operation?
A. Interchanging two rows
B. Multiplying a row by a nonzero scalar
C. Replacing a row by the sum of that row and a multiple of another row
D. Multiplying two rows together
Answer: D
Rationale: The three elementary row operations are row replacement, row scaling
(by a nonzero constant), and row interchange. Multiplying entire rows together is not
a valid elementary operation.
5. The reduced row echelon form (RREF) of a matrix is unique because:
A. The algorithm always produces the same result regardless of row operation order.
B. All matrices can be reduced to the identity matrix.
C. Row operations are commutative.
D. The determinant determines the RREF.
Answer: A
Rationale: The RREF is unique—a theorem in linear algebra states that every matrix
is row equivalent to exactly one reduced row echelon form, regardless of the
sequence of row operations used.
6. For a system Ax = b, a particular solution x_p plus any vector from the null
space of A produces:
A. A particular solution
B. The general solution
C. The trivial solution
D. An inconsistent system
Answer: B
Rationale: If x_p is one solution to Ax = b and v is any solution to Ax = 0 (the
homogeneous system), then x_p + v is also a solution. The set of all solutions is x_p +
Null(A).
7. The homogeneous system Ax = 0 always has:
, A. No solution
B. The trivial solution x = 0
C. Infinitely many solutions
D. A unique nontrivial solution
Answer: B
Rationale: Setting all variables to zero always satisfies Ax = 0. A nontrivial solution
exists if and only if there is at least one free variable.
8. If a system has more variables than equations, then:
A. The system is always inconsistent.
B. The system has a unique solution.
C. The system has either no solution or infinitely many solutions.
D. The system is always consistent.
Answer: C
Rationale: With more variables than equations, there cannot be a unique solution.
The system is either inconsistent (no solution) or consistent with at least one free
variable (infinitely many solutions).
9. A pivot position in a matrix is:
A. Any nonzero entry
B. A location that corresponds to a leading 1 in RREF
C. Always in the last column
D. The largest entry in a row
Answer: B
Rationale: A pivot position is a location in the matrix that corresponds to a leading
entry (pivot) in the row echelon form or RREF. Pivot columns are columns containing
pivot positions.
10. The dimension of the solution set of a consistent system equals:
A. The number of pivot columns
B. The number of free variables
C. The number of equations
D. The rank of the coefficient matrix
Answer: B
Rationale: If a consistent system has n variables and p pivot columns, there are n − p
free variables. The solution set is a translated subspace whose dimension equals the
number of free variables.
11. In the equation Ax = b, the matrix A represents:
