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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

sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
Thanks for this helpfull app
Axmed Reply
secA+tanA=2√5,sinA=?
richa Reply
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
NAVJIT Reply
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT
0.037 than find sin and tan?
Jon Reply
cos24/25 then find sin and tan
Deepak 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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