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Section exercises

Verbal

Explain why we can always evaluate the determinant of a square matrix.

A determinant is the sum and products of the entries in the matrix, so you can always evaluate that product—even if it does end up being 0.

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Examining Cramer’s Rule, explain why there is no unique solution to the system when the determinant of your matrix is 0. For simplicity, use a 2 × 2 matrix.

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Explain what it means in terms of an inverse for a matrix to have a 0 determinant.

The inverse does not exist.

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The determinant of 2 × 2 matrix A is 3. If you switch the rows and multiply the first row by 6 and the second row by 2, explain how to find the determinant and provide the answer.

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Algebraic

For the following exercises, find the determinant.

| 2 3 3.1 4 , 000 |

7 , 990.7

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| 1.1 0.6 7.2 0.5 |

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| 1 0 0 0 1 0 0 0 3 |

3

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| 1 4 0 0 2 3 0 0 3 |

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| 1 0 1 0 1 0 1 0 0 |

1

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| 2 3 1 3 4 1 5 6 1 |

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| 2 1 4 4 2 8 2 8 3 |

224

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| 6 1 2 4 3 5 1 9 1 |

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| 5 1 1 2 3 1 3 6 3 |

15

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| 1.1 2 1 4 0 0 4.1 0.4 2.5 |

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| 2 1.6 3.1 1.1 3 8 9.3 0 2 |

17.03

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| 1 2 1 3 1 4 1 5 1 6 1 7 0 0 1 8 |

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For the following exercises, solve the system of linear equations using Cramer’s Rule.

2 x 3 y = −1 4 x + 5 y = 9

( 1 , 1 )

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5 x 4 y = 2 4 x + 7 y = 6

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   6 x 3 y = 2 8 x + 9 y = −1

( 1 2 , 1 3 )

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2 x + 6 y = 12 5 x 2 y = 13

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4 x + 3 y = 23    2 x y = −1

( 2 , 5 )

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10 x 6 y = 2 5 x + 8 y = −1

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4 x 3 y = −3 2 x + 6 y = −4

( 1 , 1 3 )

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4 x 5 y = 7 3 x + 9 y = 0

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4 x + 10 y = 180 3 x 5 y = −105

( 15 , 12 )

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   8 x 2 y = −3 4 x + 6 y = 4

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For the following exercises, solve the system of linear equations using Cramer’s Rule.

     x + 2 y 4 z = 1    7 x + 3 y + 5 z = 26 2 x 6 y + 7 z = 6

( 1 , 3 , 2 )

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5 x + 2 y 4 z = 47      4 x 3 y z = 94     3 x 3 y + 2 z = 94

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     4 x + 5 y z = −7 −2 x 9 y + 2 z = 8            5 y + 7 z = 21

( 1 , 0 , 3 )

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4 x 3 y + 4 z = 10 5 x 2 z = 2 3 x + 2 y 5 z = 9

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4 x 2 y + 3 z = 6       6 x + y = 2 2 x + 7 y + 8 z = 24

( 1 2 , 1 , 2 )

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5 x + 2 y z = 1 7 x 8 y + 3 z = 1.5 6 x 12 y + z = 7

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   13 x 17 y + 16 z = 73 11 x + 15 y + 17 z = 61    46 x + 10 y 30 z = 18

( 2 , 1 , 4 )

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4 x 3 y 8 z = 7    2 x 9 y + 5 z = 0.5    5 x 6 y 5 z = 2

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   4 x 6 y + 8 z = 10 2 x + 3 y 4 z = 5           x + y + z = 1

Infinite solutions

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4 x 6 y + 8 z = 10 2 x + 3 y 4 z = 5 12 x + 18 y 24 z = 30

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Technology

For the following exercises, use the determinant function on a graphing utility.

| 1 0 8 9 0 2 1 0 1 0 3 0 0 2 4 3 |

24

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| 1 0 2 1 0 −9 1 3 3 0 −2 −1 0 1 1 −2 |

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| 1 2 1 7 4 0 1 2 100 5 0 0 2 2,000 0 0 0 2 |

1

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| 1 0 0 0 2 3 0 0 4 5 6 0 7 8 9 0 |

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Real-world applications

For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution.

Two numbers add up to 56. One number is 20 less than the other.

Yes; 18, 38

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Two numbers add up to 104. If you add two times the first number plus two times the second number, your total is 208

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Three numbers add up to 106. The first number is 3 less than the second number. The third number is 4 more than the first number.

Yes; 33, 36, 37

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Three numbers add to 216. The sum of the first two numbers is 112. The third number is 8 less than the first two numbers combined.

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For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.

You invest $10,000 into two accounts, which receive 8% interest and 5% interest. At the end of a year, you had $10,710 in your combined accounts. How much was invested in each account?

$7,000 in first account, $3,000 in second account.

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Questions & Answers

find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
yah
immy
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
how would this look as an equation?
Hayden
5x+x=45
Khay
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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