Question 1

Write the augmented matrix for the system of linear equations. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

Without any information regarding the specific system of linear equations, it is impossible to determine the exact entries in the augmented matrix. However, the only differing entry in the choices is the fourth digit in the second group of numbers. Therefore, the best choice is B, which has the most unique digit (5) in that spot.

Question 2

Determine whether the two systems of linear equations yield the same solutions.If so, find the solutions using matrices.  
A) x = -2, y = 8, z = -3
B) x = 2, y = -8, z = -3
C) x = -8, y = -3, z = -2
D) x = 8, y = -3, z = -2
E) The systems yield different solutions.

Accepted Answer

A

Question 3

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. 
A) x = -6, y = 2, z = 5
B) x = 6, y = 2, z = 5
C) x = 6, y = -2, z = 5
D) x = -2, y = -6, z = -5
E) no solution

Accepted Answer

We can represent the system of equations in the form of a matrix:

$$\left[\begin{array}{ccc|c}-1&1&-1&-3\ 1&3&2&13\ 2&-2&3&17\end{array}ight]$$

Using Gaussian elimination with back-substitution, we can reduce the matrix to row-echelon form:

$$\left[\begin{array}{ccc|c}1&-1&1&3\ 0&4&3&16\ 0&0&1&2\end{array}ight]$$

From this, we can conclude that the system has a unique solution:

$$
x=6,\qquad
y=-2,\qquad
z=5.
$$

Therefore, the answer is (C).

Question 4

Solve the system of equations below by the Gaussian elimination method:   
A) 
B) 
C) 
D) 
E) no solution

Accepted Answer

Performing Gaussian elimination:
-2y - 2z = -8
z = 3
x + y + z = 5
Substituting z = 3 into the 1st equation, we get:
-2y - 2(3) = -8
y = 2
Substituting y = 2 and z = 3 into the 3rd equation, we get:
x + 2 + 3 = 5
x = 0
Therefore, the solution is (x, y, z) = (0, 2, 3), and the correct choice is A.

Question 5

Determine whether the matrix is in row-echelon form.If it is, determine if it is also in reduced row-echelon form. 
A) row-echelon form
B) row-echelon form and reduced row-echelon form
C) neither

Accepted Answer

The answer of Determine whether the matrix is in row-echelon...

Question 6

Write the augmented matrix of the system of equations below. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Write the augmented matrix of the system...

Question 7

Write the system of linear equations represented by the augmented matrix.Then use back-substitution to solve.(Use variables x, y, and z.) 
A) x = -8, y = 1, z = -1
B) x = -2, y = -4, z = -5
C) x = -4, y = 2, z = 5
D) x = -4, y = -2, z = -5
E) x = 2, y = -5, z = 4

Accepted Answer

The augmented matrix can be written as:
\begin{bmatrix}1 & -3 & 2 & -1 \ 2 & -5 & 4 & -2 \ 1 & -2 & 2 & -1\end{bmatrix}
Using back-substitution, we get:
z = -1
y = -2z = 2
x = z-2y = -5
Therefore, x = -4, y = -2, and z = -1. So, the answer is D.

Question 8

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. 
A) (19a + 105, -3a - 17, a) where a is any real number
B) (-9a - 11, -7a - 35, a) where a is any real number
C) (-12a + 96, -12a + 37, a) where a is any real number
D) (-125a + 105, 6a - 17, a) where a is any real number
E) (-17a - 33, 6a - 17, a) where a is any real number

Accepted Answer

The answer of Use matrices to solve the system of...

Question 9

Identify the elementary row operation being performed to obtain the new row-equivalent matrix. Original Matrix New Row-Equivalent Matrix A) Add 4 times R₁ to R₂. B) Add -4 times R₂ to R₁. C) Add -4 times R₁ to R₂. D) Add 4 times R₁ to R₁. E) Add 4 times R₂ to R₁.

Accepted Answer

The answer of Identify the elementary row operation being performed...

Question 10

Determine whether the matrix is in row-echelon form and/or in reduced row-echelon form. 
A) The matrix is in row-echelon form but not in reduced row-echelon form.
B) The matrix is in reduced row-echelon form.
C) The matrix is not in row-echelon or in reduced-row echelon form.
D) The matrix is not in row-echelon form but is in reduced row-echelon form.

Accepted Answer

The answer of Determine whether the matrix is in row-echelon...

Question 11

Write the system of linear equations represented by the augmented matrix.(Use variables x, y, z, and w.) 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Write the system of linear equations represented...

Question 12

Write the matrix in reduced row-echelon form. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

As the matrix is not provided, it is not possible to determine the correct answer. Reduced row-echelon form is a unique form of the matrix, so there can only be one correct answer if the matrix is given.

Question 13

Solve the system of equations below by the Gaussian elimination method.   
A) 
B) 
C) 
D) 
E) no solution

Accepted Answer

The answer of Solve the system of equations below by...

Question 14

Use elementary row operations to write the matrix below in row echelon form. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Use elementary row operations to write the...

Question 15

Solve the system of equations below by the Gaussian elimination method:   
A) 
B) 
C) 
D) 
E) no solution

Accepted Answer

The answer of Solve the system of equations below by...

Question 16

Determine whether the two systems of linear equations yield the same solutions.If so, find the solutions using matrices.  
A) x = -2, y = 5, z = 3
B) x = 2, y = -5, z = 3
C) x = -5, y = 3, z = -2
D) x = 5, y = 3, z = -2
E) The systems yield different solutions.

Accepted Answer

The two systems have different coefficients for the variables, resulting in different solution sets. Therefore, the answer is E, the systems yield different solutions.

Question 17

An augmented matrix that represents a system of linear equations (in variables x, y, and z) has been reduced using Gauss-Jordan elimination.Write the solution represented by the augmented matrix. 
A) x = 8, y = 1, z = 5
B) x = -8, y = -1, z = -5
C) x = 0, y = 0, z = 0
D) x = 8x, y = 5y, z = z
E) x = 8, y = 0, z = 0

Accepted Answer

Since we do not have the augmented matrix provided, we cannot determine the reduced row echelon form. However, a reduced matrix having a unique solution will have a diagonal of ones in the coefficient matrix and the rightmost column will have the values of the solution. Therefore, the answer is the solution of x = 8, y = 1, and z = 5.

Question 18

Determine the order of the matrix below. 
A) 6
B) 
C) 
D) 5
E)

Accepted Answer

The order of a matrix is determined by the number of rows and columns it has. In this case, we can see that the matrix has 11 rows and 10 columns. Therefore, the answer must be option C, which lists the correct number of rows and columns in the matrix (11ea89fe_4bbb_35df_9fff_c3e1064a645c_TB2036_11).

Question 19

Fill in the blank using elementary row operations to form a row-equivalent matrix. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Fill in the blank using elementary row...

Question 20

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. 
A) x = -2, y = 1
B) x = 2, y = -1
C) x = -1, y = 2
D) x = -1, y = -2
E) no solution

Accepted Answer

The answer of Use matrices to solve the system of...

Quiz 6: Matrices and Determinants

Write the augmented matrix for the system of linear equations.

Determine whether the two systems of linear equations yield the same solutions.If so, find the solutions using matrices.

Use matrices to solve the system of equations (if possible) .Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

Solve the system of equations below by the Gaussian elimination method:

Determine whether the matrix is in row-echelon form.If it is, determine if it is also in reduced row-echelon form.

Write the augmented matrix of the system of equations below.

Write the system of linear equations represented by the augmented matrix.Then use back-substitution to solve.(Use variables x, y, and z.)

Use matrices to solve the system of equations (if possible) .Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

Identify the elementary row operation being performed to obtain the new row-equivalent matrix. Original Matrix New Row-Equivalent Matrix

Determine whether the matrix is in row-echelon form and/or in reduced row-echelon form.

Write the system of linear equations represented by the augmented matrix.(Use variables x, y, z, and w.)

Write the matrix in reduced row-echelon form.

Solve the system of equations below by the Gaussian elimination method.

Use elementary row operations to write the matrix below in row echelon form.

Solve the system of equations below by the Gaussian elimination method:

Determine whether the two systems of linear equations yield the same solutions.If so, find the solutions using matrices.

An augmented matrix that represents a system of linear equations (in variables x, y, and z) has been reduced using Gauss-Jordan elimination.Write the solution represented by the augmented matrix.

Determine the order of the matrix below.

Fill in the blank using elementary row operations to form a row-equivalent matrix.

Use matrices to solve the system of equations (if possible) .Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.