LINEAR ALGEBRA MATHEMATICS PREMIUM
EXAM GUIDE UPDATE 2024-2025|BRAND
NEW EXAM QUESTIONS AND CORRECT
ANSWERS,ALL GRADED A+|GUARANTEED
SUCCESS|LATEST UPDATE
1.1
10. a.
Is the statement "Every elementary row operation is reversible" true or false?
Explain. - ANSWER-✔True, because replacement, interchanging, and scaling are
all reversible.
1.1
10. b.
Is the statement A 5×6 matrix has six rows" true or false? Explain. - ANSWER-
✔False, because a 5×6 matrix has five rows and six columns.
1.1
10. c.
Is the statement "The solution set of a linear system involving variables x1, ..., xn is a
list of numbers (s1, ..., sn) that makes each equation in the system a true statement
when the values s1, ..., sn are substituted for x1, ..., xn, respectively" true or false?
Explain. - ANSWER-✔False, because the description applies to a single solution. The
solution set consists of all possible solutions.
,1.1
10. d.
Is the statement "Two fundamental questions about a linear system involve
existence and uniqueness" true or false? Explain. - ANSWER-✔True, because
two fundamental questions address whether the solution exists and whether there is
only one solution.
1.1
11. a.
Is the statement "Two matrices are row equivalent if they have the same number
of rows" true or false? Explain. - ANSWER-✔ False, because if two matrices
are row equivalent it means that there exists a sequence of row operations that
transforms one matrix to the other.
1.1
11. b.
Is the statement "Elementary row operations on an augmented matrix never
change the solution set of the associated linear system" true or false? Explain. -
ANSWER-✔True, because the elementary row operations replace a system with an
equivalent system.
1.1
11. c.
Is the statement "Two equivalent linear systems can have different solution
sets" true or false? Explain. - ANSWER-✔False, because two systems are called
equivalent if they have the same solution set.
1.1
11. d.
Is the statement "A consistent system of linear equations has one or more
solutions" true or false? Explain. - ANSWER-✔True, a consistent system is defined
as a system that has at least one solution.
1.2
6. a.
In some cases, a matrix may be row reduced to more than one matrix in reduced
echelon form, using different sequences of row operations. - ANSWER-✔The
statement is false. Each matrix is row equivalent to one and only one reduced
echelon matrix.
,1.2
6. b.
The row reduction algorithm applies only to augmented matrices for a linear system.
- ANSWER-✔The statement is false. The algorithm applies to any matrix, whether
or not the matrix is viewed as an augmented matrix for a linear system.
1.2
6. c.
A basic variable in a linear system is a variable that corresponds to a pivot column in
the coefficient matrix. - ANSWER-✔The statement is true. It is the definition of a
basic variable.
1.2
6. d.
Finding a parametric description of the solution set of a linear system is the same as
solving the system. - ANSWER-✔The statement is false. The solution set of a linear
system can only be expressed using a parametric description if the system has at
least one solution.
1.2
6. e.
If one row in an echelon form of an augmented matrix is left bracket
[0 0 0 5 0 ]
then the associated linear system is inconsistent. - ANSWER-✔The statement is false.
The indicated row corresponds to the equation
5x(subscript 4) = 0,
which does not by itself make the system inconsistent.
1.2
7. Part 1
Suppose the coefficient matrix of a linear system of four equations in four variables
has a pivot in each column. Explain why the system has a unique solution.
What must be true of a linear system for it to have a unique solution? Select all
that apply. - ANSWER-✔The system is consistent.
The system has no free variables.
1.2
7. Part 2
Use the given assumption that the coefficient matrix of the linear system of four
equations in four variables has a pivot in each column to determine the dimensions
of the coefficient matrix. - ANSWER-✔The coefficient matrix has
, four rows and four columns.
1.2
7. Part 3
Let the coefficient matrix be in reduced echelon form with a pivot in each column,
since each matrix is equivalent to one and only one reduced echelon matrix.
Construct a matrix with the dimensions determined in the previous step that is in
reduced echelon form and has a pivot in each column. - ANSWER-✔[ 1 0 0 0 ]
[0100]
[0010]
[0001]
1.2
7. Part 4
Now find an augmented matrix in reduced echelon form that represents a linear
system of four equations in four variables for which the corresponding coefficient
matrix has a pivot in each column. Choose the correct answer below. - ANSWER-
✔[ 1 0 0 0 a ]
[0100b]
[0010c]
[0001d]
1.2
7. Part 5
Use the augmented matrix to determine if the linear system is consistent. Is the
linear system represented by the augmented matrix consistent? - ANSWER-
✔ Yes, because the rightmost column of the augmented matrix is not a pivot
column.
1.2
7. Part 6
Write the system of equations corresponding to the augmented matrix found above
to determine the number of free variables. - ANSWER-✔x1 = a
x2 = b
x3 = c
x4 = d
1.2
7. Part 7
Free variables are variables that can take on any value. How many free variables are
in the system? - ANSWER-✔None, because x1, x2, x3, and x4 are all fixed values
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