Question 1

Use determinants to find the area of the triangle with vertices at the given points. ​
P(0,0),Q(4,0),R(4,3)
​

Accepted Answer

A) A = 12 
B) A = 4 
C) A = 6 
D) A = 3 
E) none of these 
A) A = 12 
B) A = 4 
C) A = 6 
D) A = 3 
E) none of these C

Question 2

Use a determinant and the given vertices of a triangle to find the area of the triangle. ?
(-2,5), (3,3), (-1,6)
?

Accepted Answer

A)  $\frac { 7 } { 2 }$ 
B)  $\frac { 2 } { 19 }$ 
C)  $\frac { 3 } { 7 }$ 
D)  $\frac { 6 } { 11 }$ 
E)  $\frac { 5 } { 17 }$ 
A)  $\frac { 7 } { 2 }$ 
B)  $\frac { 2 } { 19 }$ 
C)  $\frac { 3 } { 7 }$ 
D)  $\frac { 6 } { 11 }$ 
E)  $\frac { 5 } { 17 }$ A

Question 3

Use a determinant and the given vertices of a triangle to find the area of the triangle. ?
(-5,5), (3,4), (4,-6)
?

Accepted Answer

A)  $\frac { 79 } { 2 }$ 
B) - $\frac { 79 } { 2 }$ 
C)  $\frac { 4 } { 19 }$ 
D) 79 
E)  $\frac { 4 } { 11 }$ 
A)  $\frac { 79 } { 2 }$ 
B) - $\frac { 79 } { 2 }$ 
C)  $\frac { 4 } { 19 }$ 
D) 79 
E)  $\frac { 4 } { 11 }$ A

Question 4

Determine a positive value for y such that a triangle with vertices (-4,2), (6,4)and (0,y)has an area of 16 square units.

Accepted Answer

A) 11 
B) 8 
C) 6 
D) 9 
E) 1 
A) 11 
B) 8 
C) 6 
D) 9 
E) 1

Question 5

Find the uncoded 1 × 3 row matrices for the message "TWO IF BY LAND";then encode the message using the encoding matrix $$\left[ \begin{array} { c c c } 
0 & - 1 & 1 \
1 & - 3 & 3 \
- 3 & - 1 & 0
\end{array} \right]$$ .Show all your work.

Accepted Answer

The answer of Find the uncoded 1 × 3 row...

Question 6

Use Cramer's Rule to solve the following system of linear equations: $$\left\{ \begin{aligned}
14 x - 21 y + 14 z & = 1 \
- 21 x + 14 y + 7 z & = 5 \
28 x + 7 y - 21 z & = 5
\end{aligned} \right.$$

Accepted Answer

A)  $$x = \frac { 5 } { 7 } ; y = \frac { 55 } { 49 } ; z = \frac { 95 } { 147 }$$ 
B)  $$x = \frac { 3 } { 7 } ; y = \frac { 45 } { 98 } ; z = \frac { 12 } { 49 }$$ 
C)  $$x = \frac { 11 } { 14 } ; y = \frac { 115 } { 98 } ; z = \frac { 106 } { 147 }$$ 
D)  $$x = \frac { 11 } { 21 } ; y = \frac { 115 } { 147 } ; z = \frac { 106 } { 147 }$$ 
E)  $$x = \frac { 3 } { 7 } ; y = \frac { 75 } { 98 } ; z = \frac { 27 } { 49 }$$ 
A)  $$x = \frac { 5 } { 7 } ; y = \frac { 55 } { 49 } ; z = \frac { 95 } { 147 }$$ 
B)  $$x = \frac { 3 } { 7 } ; y = \frac { 45 } { 98 } ; z = \frac { 12 } { 49 }$$ 
C)  $$x = \frac { 11 } { 14 } ; y = \frac { 115 } { 98 } ; z = \frac { 106 } { 147 }$$ 
D)  $$x = \frac { 11 } { 21 } ; y = \frac { 115 } { 147 } ; z = \frac { 106 } { 147 }$$ 
E)  $$x = \frac { 3 } { 7 } ; y = \frac { 75 } { 98 } ; z = \frac { 27 } { 49 }$$

Question 7

Use a determinant to find an equation of the line passing through the points. ​
(-5,3), (1,1)
​

Accepted Answer

A) 2x -6y + 8 = 0 
B) 2x + 6y + 8 = 0 
C) 2x -6y - 8 = 0 
D) 2x + 6y - 8 = 0 
E) 2x + 6y - 8 = 5 
A) 2x -6y + 8 = 0 
B) 2x + 6y + 8 = 0 
C) 2x -6y - 8 = 0 
D) 2x + 6y - 8 = 0 
E) 2x + 6y - 8 = 5

Question 8

Use a determinant and the given vertices of a triangle to find the area of the triangle.?   ? ?

Accepted Answer

A)  $\frac { 41 } { 2 }$ 
B) 21 
C)  $\frac { 39 } { 2 }$ 
D) 20 
E)  $\frac { 43 } { 2 }$ 
A)  $\frac { 41 } { 2 }$ 
B) 21 
C)  $\frac { 39 } { 2 }$ 
D) 20 
E)  $\frac { 43 } { 2 }$

Question 9

Use Cramer's Rule to solve the following system of linear equations:? $$\left\{ \begin{aligned}
3 x &&- 6 z & = 8 \
&&- 6 y + 13 z & = 3 \
6 x & &+15 z & = 0
\end{aligned} \right.$$

Accepted Answer

A) ?x = $\frac { 40 } { 27 }$ ,y = $- \frac { 16 } { 27 }$ ,z = $- \frac { 289 } { 162 }$ 
B) ??x = $\frac { 40 } { 27 }$ ,y = $- \frac { 289 } { 162 }$ ,z = $- \frac { 16 } { 27 }$ ? 
C) ?x = $\frac { 27 } { 40 }$ ,y = $- \frac { 162 } { 289 }$ ,z = $- \frac { 27 } { 16 }$ 
D) ?x = $\frac { 27 } { 40 }$ ,y = $- \frac { 289 } { 162 }$ ,z = $- \frac { 16 } { 27 }$ 
E) ?x = $\frac { 40 } { 27 }$ ,y = $- \frac { 162 } { 289 }$ ,z = $- \frac { 16 } { 27 }$ 
A) ?x = $\frac { 40 } { 27 }$ ,y = $- \frac { 16 } { 27 }$ ,z = $- \frac { 289 } { 162 }$ 
B) ??x = $\frac { 40 } { 27 }$ ,y = $- \frac { 289 } { 162 }$ ,z = $- \frac { 16 } { 27 }$ ? 
C) ?x = $\frac { 27 } { 40 }$ ,y = $- \frac { 162 } { 289 }$ ,z = $- \frac { 27 } { 16 }$ 
D) ?x = $\frac { 27 } { 40 }$ ,y = $- \frac { 289 } { 162 }$ ,z = $- \frac { 16 } { 27 }$ 
E) ?x = $\frac { 40 } { 27 }$ ,y = $- \frac { 162 } { 289 }$ ,z = $- \frac { 16 } { 27 }$

Question 10

Use Cramer's Rule to solve (if possible)the system of equations.? $$\left\{ \begin{array} { l } 
x + 2 y + 3 z = - 5 \
- 2 x + y - z = 10 \
3 x - 3 y + 2 z = - 21
\end{array} \right.$$ ?

Accepted Answer

A) (-2,3,-3) 
B) (7,-3,2) 
C) (2,3,-3) 
D) (2,-3,-3) 
E) (-2,-3,-3) 
A) (-2,3,-3) 
B) (7,-3,2) 
C) (2,3,-3) 
D) (2,-3,-3) 
E) (-2,-3,-3)

Question 11

Use a determinant and the given vertices of a triangle to find the area of the triangle. ?
(0,-3), (-3,3), (4,4)
?

Accepted Answer

A)  $\frac { 4 } { 7 }$ 
B)  $\frac { 5 } { 11 }$ 
C)  $\frac { 45 } { 2 }$ 
D)  $\frac { 4 } { 19 }$ 
E) 45 
A)  $\frac { 4 } { 7 }$ 
B)  $\frac { 5 } { 11 }$ 
C)  $\frac { 45 } { 2 }$ 
D)  $\frac { 4 } { 19 }$ 
E) 45

Question 12

Use Cramer's Rule to solve (if possible)the system of equations. ?? $$\left\{ \begin{array} { r } 
5 x - 4 y + z = - 12 \
- x + 2 y - 2 z = 12 \
3 x + y + z = - 2
\end{array} \right.$$ ?

Accepted Answer

A) (0,2,-4) 
B) (0,-2,4) 
C) (0,-2,-4) 
D) (0,2,0) 
E) (0,2,4) 
A) (0,2,-4) 
B) (0,-2,4) 
C) (0,-2,-4) 
D) (0,2,0) 
E) (0,2,4)

Question 13

Use a determinant to find an equation of the line passing through the points. ​
(0,0), (-8,2)
​

Accepted Answer

A) 2x + 8y = 0 
B) 8x - 2y = 2 
C) 2x + 8y = 8 
D) 8x - 2y = 0 
E) 2x - 8y = 0 
A) 2x + 8y = 0 
B) 8x - 2y = 2 
C) 2x + 8y = 8 
D) 8x - 2y = 0 
E) 2x - 8y = 0

Question 14

Use a determinant and the given vertices of a triangle to find the area of the triangle.?   ?

Accepted Answer

A) 27 
B)  $\frac { 4 } { 19 }$ 
C) 29 
D)  $\frac { 25 } { 2 }$ 
E)  $\frac { 6 } { 11 }$ 
A) 27 
B)  $\frac { 4 } { 19 }$ 
C) 29 
D)  $\frac { 25 } { 2 }$ 
E)  $\frac { 6 } { 11 }$

Question 15

Use a determinant to find the area of the triangle shown below.

Accepted Answer

A) 14 square units 
B)  $\frac { 57 } { 4 }$ square units 
C)  $\frac { 27 } { 2 }$ square units 
D) 12 square units 
E)  $\frac { 85 } { 6 }$ square units 
A) 14 square units 
B)  $\frac { 57 } { 4 }$ square units 
C)  $\frac { 27 } { 2 }$ square units 
D) 12 square units 
E)  $\frac { 85 } { 6 }$ square units

Question 16

Use a determinant to find an equation of the line passing through the points. ​
(8,8), (-4,-8)
​

Accepted Answer

A) 4x + 3y + 8 = 0 
B) 4x + 3y - 8 = 0 
C) 4x - 3y + 8 = 0 
D) 4x - 3y - 8 = 0 
E) 4x - 3y = 0 
A) 4x + 3y + 8 = 0 
B) 4x + 3y - 8 = 0 
C) 4x - 3y + 8 = 0 
D) 4x - 3y - 8 = 0 
E) 4x - 3y = 0

Question 17

Use Cramer's Rule to solve (if possible)the system of equations.? $$\left\{ \begin{array} { r } 
- 0.4 x + 0.8 y = 5.52 \
0.2 x + 0.3 y = 2.63
\end{array} \right.$$ ?

Accepted Answer

A)  $$\left( \frac { 77 } { 10 } , \frac { 8 } { 5 } \right)$$ 
B)  $$\left( \frac { 8 } { 5 } , \frac { 77 } { 10 } \right)$$ 
C)  $$\left( - \frac { 8 } { 5 } , - \frac { 77 } { 10 } \right)$$ 
D)  $$\left( \frac { 8 } { 5 } , - \frac { 77 } { 10 } \right)$$ 
E)  $$\left( - \frac { 8 } { 5 } , \frac { 77 } { 10 } \right)$$ 
A)  $$\left( \frac { 77 } { 10 } , \frac { 8 } { 5 } \right)$$ 
B)  $$\left( \frac { 8 } { 5 } , \frac { 77 } { 10 } \right)$$ 
C)  $$\left( - \frac { 8 } { 5 } , - \frac { 77 } { 10 } \right)$$ 
D)  $$\left( \frac { 8 } { 5 } , - \frac { 77 } { 10 } \right)$$ 
E)  $$\left( - \frac { 8 } { 5 } , \frac { 77 } { 10 } \right)$$

Question 18

Use a determinant to find an equation of the line passing through the points (-5,-1)and (2,4). ​

Accepted Answer

A) 5x + 7y - 18 = 0 
B) -5x - 7y - 18 = 0 
C) -5x + 7y + 18 = 0 
D) 5x - 7y - 18 = 0 
E) -5x + 7y - 18 = 0 
A) 5x + 7y - 18 = 0 
B) -5x - 7y - 18 = 0 
C) -5x + 7y + 18 = 0 
D) 5x - 7y - 18 = 0 
E) -5x + 7y - 18 = 0

Question 19

Use a determinant to determine whether the points (-3,-9), (-5,-11)and (0,-7)are collinear. ?

Accepted Answer

A)  $$\left| \begin{array} { c c c } 
- 3 & - 9 & 1 \
- 5 & - 11 & 1 \
0 & - 7 & 1
\end{array} \right| = 2 \text {; therefore, the points are not collinear }$$ 
B)  $$\left| \begin{array} { c c c } 
- 3 & - 9 & 1 \
- 5 & - 11 & 1 \
0 & - 7 & 1
\end{array} \right| = 0 \text {; therefore, the points are collinear }$$ 
A)  $$\left| \begin{array} { c c c } 
- 3 & - 9 & 1 \
- 5 & - 11 & 1 \
0 & - 7 & 1
\end{array} \right| = 2 \text {; therefore, the points are not collinear }$$ 
B)  $$\left| \begin{array} { c c c } 
- 3 & - 9 & 1 \
- 5 & - 11 & 1 \
0 & - 7 & 1
\end{array} \right| = 0 \text {; therefore, the points are collinear }$$

Question 20

Use a determinant to find y such that (8,-20), (16,y),and (20,-8)are collinear.

Accepted Answer

A) y = -12 
B) y = 20 
C) y = -44 
D) y = 4 
E) y = -8 
A) y = -12 
B) y = 20 
C) y = -44 
D) y = 4 
E) y = -8

Exam 51: Applications of Matrices and Determinants

Use determinants to find the area of the triangle with vertices at the given points. ​ P(0,0),Q(4,0),R(4,3) ​

Use a determinant and the given vertices of a triangle to find the area of the triangle. ? (-2,5), (3,3), (-1,6) ?

Use a determinant and the given vertices of a triangle to find the area of the triangle. ? (-5,5), (3,4), (4,-6) ?

Determine a positive value for y such that a triangle with vertices (-4,2), (6,4)and (0,y)has an area of 16 square units.

Find the uncoded 1 × 3 row matrices for the message "TWO IF BY LAND";then encode the message using the encoding matrix [0−111−33−3−10]\left[ \begin{array} { c c c } 0 & - 1 & 1 \\1 & - 3 & 3 \\- 3 & - 1 & 0\end{array} \right]​01−3​−1−3−1​130​​ .Show all your work.

Use Cramer's Rule to solve the following system of linear equations: {14x−21y+14z=1−21x+14y+7z=528x+7y−21z=5\left\{ \begin{aligned}14 x - 21 y + 14 z & = 1 \\- 21 x + 14 y + 7 z & = 5 \\28 x + 7 y - 21 z & = 5\end{aligned} \right.⎩⎨⎧​14x−21y+14z−21x+14y+7z28x+7y−21z​=1=5=5​

Use a determinant to find an equation of the line passing through the points. ​ (-5,3), (1,1) ​

Use a determinant and the given vertices of a triangle to find the area of the triangle.? ? ?

Use Cramer's Rule to solve the following system of linear equations:? {3x−6z=8−6y+13z=36x+15z=0\left\{ \begin{aligned}3 x &&- 6 z & = 8 \\&&- 6 y + 13 z & = 3 \\6 x & &+15 z & = 0\end{aligned} \right.⎩⎨⎧​3x6x​​−6z−6y+13z+15z​=8=3=0​

Use Cramer's Rule to solve (if possible)the system of equations.? {x+2y+3z=−5−2x+y−z=103x−3y+2z=−21\left\{ \begin{array} { l } x + 2 y + 3 z = - 5 \\- 2 x + y - z = 10 \\3 x - 3 y + 2 z = - 21\end{array} \right.⎩⎨⎧​x+2y+3z=−5−2x+y−z=103x−3y+2z=−21​ ?

Use a determinant and the given vertices of a triangle to find the area of the triangle. ? (0,-3), (-3,3), (4,4) ?

Use Cramer's Rule to solve (if possible)the system of equations. ?? {5x−4y+z=−12−x+2y−2z=123x+y+z=−2\left\{ \begin{array} { r } 5 x - 4 y + z = - 12 \\- x + 2 y - 2 z = 12 \\3 x + y + z = - 2\end{array} \right.⎩⎨⎧​5x−4y+z=−12−x+2y−2z=123x+y+z=−2​ ?

Use a determinant to find an equation of the line passing through the points. ​ (0,0), (-8,2) ​

Use a determinant and the given vertices of a triangle to find the area of the triangle.? ?

Use a determinant to find the area of the triangle shown below.

Use a determinant to find an equation of the line passing through the points. ​ (8,8), (-4,-8) ​

Use Cramer's Rule to solve (if possible)the system of equations.? {−0.4x+0.8y=5.520.2x+0.3y=2.63\left\{ \begin{array} { r } - 0.4 x + 0.8 y = 5.52 \\0.2 x + 0.3 y = 2.63\end{array} \right.{−0.4x+0.8y=5.520.2x+0.3y=2.63​ ?

Use a determinant to find an equation of the line passing through the points (-5,-1)and (2,4). ​

Use a determinant to determine whether the points (-3,-9), (-5,-11)and (0,-7)are collinear. ?

Use a determinant to find y such that (8,-20), (16,y),and (20,-8)are collinear.

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

Use determinants to find the area of the triangle with vertices at the given points. P(0,0),Q(4,0),R(4,3)

Find the uncoded 1 × 3 row matrices for the message "TWO IF BY LAND";then encode the message using the encoding matrix $\left[ \begin{array} { c c c } 0 & - 1 & 1 \\1 & - 3 & 3 \\- 3 & - 1 & 0\end{array} \right]$ .Show all your work.

Use Cramer's Rule to solve the following system of linear equations: $\left\{ \begin{aligned}14 x - 21 y + 14 z & = 1 \\- 21 x + 14 y + 7 z & = 5 \\28 x + 7 y - 21 z & = 5\end{aligned} \right.$

Use a determinant to find an equation of the line passing through the points. (-5,3), (1,1)

Use Cramer's Rule to solve the following system of linear equations:? $\left\{ \begin{aligned}3 x &&- 6 z & = 8 \\&&- 6 y + 13 z & = 3 \\6 x & &+15 z & = 0\end{aligned} \right.$

Use Cramer's Rule to solve (if possible)the system of equations.? $\left\{ \begin{array} { l } x + 2 y + 3 z = - 5 \\- 2 x + y - z = 10 \\3 x - 3 y + 2 z = - 21\end{array} \right.$ ?

Use Cramer's Rule to solve (if possible)the system of equations. ?? $\left\{ \begin{array} { r } 5 x - 4 y + z = - 12 \\- x + 2 y - 2 z = 12 \\3 x + y + z = - 2\end{array} \right.$ ?

Use a determinant to find an equation of the line passing through the points. (0,0), (-8,2)

Use a determinant to find an equation of the line passing through the points. (8,8), (-4,-8)

Use Cramer's Rule to solve (if possible)the system of equations.? $\left\{ \begin{array} { r } - 0.4 x + 0.8 y = 5.52 \\0.2 x + 0.3 y = 2.63\end{array} \right.$ ?

Use a determinant to find an equation of the line passing through the points (-5,-1)and (2,4).